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Metamath Proof Explorer

Theorem fmptf

Description: Functionality of the mapping operation. (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses fmptf.1 
|- F/_ x B
fmptf.2 
|- F = ( x e. A |-> C )
Assertion fmptf 
|- ( A. x e. A C e. B <-> F : A --> B )

Proof

Step Hyp Ref Expression
1 fmptf.1 
 |-  F/_ x B
2 fmptf.2 
 |-  F = ( x e. A |-> C )
3 nfv 
 |-  F/ y C e. B
4 nfcsb1v 
 |-  F/_ x [_ y / x ]_ C
5 4 1 nfel 
 |-  F/ x [_ y / x ]_ C e. B
6 csbeq1a 
 |-  ( x = y -> C = [_ y / x ]_ C )
7 6 eleq1d 
 |-  ( x = y -> ( C e. B <-> [_ y / x ]_ C e. B ) )
8 3 5 7 cbvralw 
 |-  ( A. x e. A C e. B <-> A. y e. A [_ y / x ]_ C e. B )
9 nfcv 
 |-  F/_ y C
10 9 4 6 cbvmpt 
 |-  ( x e. A |-> C ) = ( y e. A |-> [_ y / x ]_ C )
11 2 10 eqtri 
 |-  F = ( y e. A |-> [_ y / x ]_ C )
12 11 fmpt 
 |-  ( A. y e. A [_ y / x ]_ C e. B <-> F : A --> B )
13 8 12 bitri 
 |-  ( A. x e. A C e. B <-> F : A --> B )