Theorem tpf1ofv0

Description: The value of a one-to-one function onto a triple at 0. (Contributed by AV, 20-Jul-2025)

	Ref	Expression
Hypothesis	tpf1o.f	|- F = ( x e. ( 0 ..^ 3 ) |-> if ( x = 0 , A , if ( x = 1 , B , C ) ) )
Assertion	tpf1ofv0	|- ( A e. V -> ( F ` 0 ) = A )

Proof

Step	Hyp	Ref	Expression
1		tpf1o.f	|- F = ( x e. ( 0 ..^ 3 ) |-> if ( x = 0 , A , if ( x = 1 , B , C ) ) )
2	1	a1i	|- ( A e. V -> F = ( x e. ( 0 ..^ 3 ) |-> if ( x = 0 , A , if ( x = 1 , B , C ) ) ) )
3		iftrue	|- ( x = 0 -> if ( x = 0 , A , if ( x = 1 , B , C ) ) = A )
4	3	adantl	|- ( ( A e. V /\ x = 0 ) -> if ( x = 0 , A , if ( x = 1 , B , C ) ) = A )
5		3nn	|- 3 e. NN
6		lbfzo0	|- ( 0 e. ( 0 ..^ 3 ) <-> 3 e. NN )
7	5 6	mpbir	|- 0 e. ( 0 ..^ 3 )
8	7	a1i	|- ( A e. V -> 0 e. ( 0 ..^ 3 ) )
9		id	|- ( A e. V -> A e. V )
10	2 4 8 9	fvmptd	|- ( A e. V -> ( F ` 0 ) = A )