if f:x→x be a mapping prove that (f⊆I_X ∨I_X ⊆f)→f=I


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Question Number 109149 by mohammad17 last updated on 21/Aug/20

$i f f : x \to x b e a m a p p i n g p r o v e t h a t (f \subseteq I_{X} \lor I_{X} \subseteq f) \to f = I_{X} ?$ $h e l p m e s i r$
Commented by kaivan.ahmadi last updated on 21/Aug/20

$X \overset{f}{\to} X$ $X \overset{I}{\to} X, I (x) = x$ $i f f \subseteq I_{X} t h e n f o r e a c h x i n X w e {h a v e I}_{X} (x) = x$ $o n t h e o t h e r h a n d x = f (y) f o r s o m e y i n X$ $s o I_{X} \subseteq f a n d I_{X} = f .$ $i f I_{X} \subseteq f t h e n f o r e a c h x i n X w e h a v e f (x) \in X$ $l e t f (x) = y \in X a n d s o f (x) = y = I_{X} (y) s o f \subseteq I_{X}, h e n c e$ $f = I_{X} .$
Commented by mohammad17 last updated on 21/Aug/20

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