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Metamath Proof Explorer

Theorem constcof

Description: Composition with a constant function. See also fcoconst . (Contributed by Thierry Arnoux, 11-Jan-2026)

Ref Expression
Hypotheses constcof.1 
|- ( ph -> F : X --> I )
constcof.2 
|- ( ph -> Y e. V )
Assertion constcof 
|- ( ph -> ( ( I X. { Y } ) o. F ) = ( X X. { Y } ) )

Proof

Step Hyp Ref Expression
1 constcof.1 
 |-  ( ph -> F : X --> I )
2 constcof.2 
 |-  ( ph -> Y e. V )
3 fnconstg 
 |-  ( Y e. V -> ( I X. { Y } ) Fn I )
4 2 3 syl 
 |-  ( ph -> ( I X. { Y } ) Fn I )
5 fnfco 
 |-  ( ( ( I X. { Y } ) Fn I /\ F : X --> I ) -> ( ( I X. { Y } ) o. F ) Fn X )
6 4 1 5 syl2anc 
 |-  ( ph -> ( ( I X. { Y } ) o. F ) Fn X )
7 1 adantr 
 |-  ( ( ph /\ x e. X ) -> F : X --> I )
8 simpr 
 |-  ( ( ph /\ x e. X ) -> x e. X )
9 7 8 fvco3d 
 |-  ( ( ph /\ x e. X ) -> ( ( ( I X. { Y } ) o. F ) ` x ) = ( ( I X. { Y } ) ` ( F ` x ) ) )
10 2 adantr 
 |-  ( ( ph /\ x e. X ) -> Y e. V )
11 1 ffvelcdmda 
 |-  ( ( ph /\ x e. X ) -> ( F ` x ) e. I )
12 fvconst2g 
 |-  ( ( Y e. V /\ ( F ` x ) e. I ) -> ( ( I X. { Y } ) ` ( F ` x ) ) = Y )
13 10 11 12 syl2anc 
 |-  ( ( ph /\ x e. X ) -> ( ( I X. { Y } ) ` ( F ` x ) ) = Y )
14 9 13 eqtrd 
 |-  ( ( ph /\ x e. X ) -> ( ( ( I X. { Y } ) o. F ) ` x ) = Y )
15 6 14 fconst7v 
 |-  ( ph -> ( ( I X. { Y } ) o. F ) = ( X X. { Y } ) )