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 } ) )