Theorem bj-mptval 33070

Description: Value of a function given in maps-to notation. (Contributed by BJ, 30-Dec-2020.)

Hypothesis

Ref	Expression
bj-mptval.nf	|- F/_ x A

Assertion

Ref	Expression
bj-mptval	|- ( A. x e. A B e. V -> ( X e. A -> ( ( ( x e. A |-> B ) ` X ) = Y <-> X ( x e. A |-> B ) Y ) ) )

Proof of Theorem bj-mptval

Step	Hyp	Ref	Expression
1		bj-mptval.nf	. . 3 |- F/_ x A
2	1	fnmptf 6016	. 2 |- ( A. x e. A B e. V -> ( x e. A |-> B ) Fn A )
3		fnbrfvb 6236	. . 3 |- ( ( ( x e. A |-> B ) Fn A /\ X e. A ) -> ( ( ( x e. A |-> B ) ` X ) = Y <-> X ( x e. A |-> B ) Y ) )
4	3	ex 450	. 2 |- ( ( x e. A |-> B ) Fn A -> ( X e. A -> ( ( ( x e. A |-> B ) ` X ) = Y <-> X ( x e. A |-> B ) Y ) ) )
5	2, 4	syl 17	1 |- ( A. x e. A B e. V -> ( X e. A -> ( ( ( x e. A |-> B ) ` X ) = Y <-> X ( x e. A |-> B ) Y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   F/_wnfc 2751   A.wral 2912   class class class wbr 4653   |-> cmpt 4729   Fn wfn 5883   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by: (None)