Theorem abfmpeld 29454

Description: Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)

Hypotheses

Ref	Expression
abfmpeld.1	|- F = ( x e. V |-> { y | ps } )
abfmpeld.2	|- ( ph -> { y | ps } e. _V )
abfmpeld.3	|- ( ph -> ( ( x = A /\ y = B ) -> ( ps <-> ch ) ) )

Assertion

Ref	Expression
abfmpeld	|- ( ph -> ( ( A e. V /\ B e. W ) -> ( B e. ( F ` A ) <-> ch ) ) )

Distinct variable groups:   x, y, A   x, B, y   x, F, y   x, V, y   y, W   ch, x, y   ph, x, y

Allowed substitution hints:   ps( x, y)   W( x)

Proof of Theorem abfmpeld

Step	Hyp	Ref	Expression
1		abfmpeld.2	. . . . . . . . . 10 |- ( ph -> { y | ps } e. _V )
2	1	alrimiv 1855	. . . . . . . . 9 |- ( ph -> A. x { y | ps } e. _V )
3		csbexg 4792	. . . . . . . . 9 |- ( A. x { y | ps } e. _V -> [_ A / x ]_ { y | ps } e. _V )
4	2, 3	syl 17	. . . . . . . 8 |- ( ph -> [_ A / x ]_ { y | ps } e. _V )
5		abfmpeld.1	. . . . . . . . 9 |- F = ( x e. V |-> { y | ps } )
6	5	fvmpts 6285	. . . . . . . 8 |- ( ( A e. V /\ [_ A / x ]_ { y | ps } e. _V ) -> ( F ` A ) = [_ A / x ]_ { y | ps } )
7	4, 6	sylan2 491	. . . . . . 7 |- ( ( A e. V /\ ph ) -> ( F ` A ) = [_ A / x ]_ { y | ps } )
8		csbab 4008	. . . . . . 7 |- [_ A / x ]_ { y | ps } = { y | [. A / x ]. ps }
9	7, 8	syl6eq 2672	. . . . . 6 |- ( ( A e. V /\ ph ) -> ( F ` A ) = { y | [. A / x ]. ps } )
10	9	eleq2d 2687	. . . . 5 |- ( ( A e. V /\ ph ) -> ( B e. ( F ` A ) <-> B e. { y | [. A / x ]. ps } ) )
11	10	adantl 482	. . . 4 |- ( ( B e. W /\ ( A e. V /\ ph ) ) -> ( B e. ( F ` A ) <-> B e. { y | [. A / x ]. ps } ) )
12		simpll 790	. . . . . . . 8 |- ( ( ( A e. V /\ ph ) /\ y = B ) -> A e. V )
13		abfmpeld.3	. . . . . . . . . . 11 |- ( ph -> ( ( x = A /\ y = B ) -> ( ps <-> ch ) ) )
14	13	ancomsd 470	. . . . . . . . . 10 |- ( ph -> ( ( y = B /\ x = A ) -> ( ps <-> ch ) ) )
15	14	adantl 482	. . . . . . . . 9 |- ( ( A e. V /\ ph ) -> ( ( y = B /\ x = A ) -> ( ps <-> ch ) ) )
16	15	impl 650	. . . . . . . 8 |- ( ( ( ( A e. V /\ ph ) /\ y = B ) /\ x = A ) -> ( ps <-> ch ) )
17	12, 16	sbcied 3472	. . . . . . 7 |- ( ( ( A e. V /\ ph ) /\ y = B ) -> ( [. A / x ]. ps <-> ch ) )
18	17	ex 450	. . . . . 6 |- ( ( A e. V /\ ph ) -> ( y = B -> ( [. A / x ]. ps <-> ch ) ) )
19	18	alrimiv 1855	. . . . 5 |- ( ( A e. V /\ ph ) -> A. y ( y = B -> ( [. A / x ]. ps <-> ch ) ) )
20		elabgt 3347	. . . . 5 |- ( ( B e. W /\ A. y ( y = B -> ( [. A / x ]. ps <-> ch ) ) ) -> ( B e. { y | [. A / x ]. ps } <-> ch ) )
21	19, 20	sylan2 491	. . . 4 |- ( ( B e. W /\ ( A e. V /\ ph ) ) -> ( B e. { y | [. A / x ]. ps } <-> ch ) )
22	11, 21	bitrd 268	. . 3 |- ( ( B e. W /\ ( A e. V /\ ph ) ) -> ( B e. ( F ` A ) <-> ch ) )
23	22	an13s 845	. 2 |- ( ( ph /\ ( A e. V /\ B e. W ) ) -> ( B e. ( F ` A ) <-> ch ) )
24	23	ex 450	1 |- ( ph -> ( ( A e. V /\ B e. W ) -> ( B e. ( F ` A ) <-> ch ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   [.wsbc 3435   [_csb 3533   |-> cmpt 4729   ` 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-fal 1489  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-csb 3534  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-fv 5896

This theorem is referenced by: (None)