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Metamath Proof Explorer

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 )