Theorem abfmpel 29455

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

Hypotheses

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

Assertion

Ref	Expression
abfmpel	|- ( ( A e. V /\ B e. W ) -> ( B e. ( F ` A ) <-> ps ) )

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

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

Proof of Theorem abfmpel

Step	Hyp	Ref	Expression
1		abfmpel.2	. . . . . . 7 |- { y | ph } e. _V
2	1	csbex 4793	. . . . . 6 |- [_ A / x ]_ { y | ph } e. _V
3		abfmpel.1	. . . . . . 7 |- F = ( x e. V |-> { y | ph } )
4	3	fvmpts 6285	. . . . . 6 |- ( ( A e. V /\ [_ A / x ]_ { y | ph } e. _V ) -> ( F ` A ) = [_ A / x ]_ { y | ph } )
5	2, 4	mpan2 707	. . . . 5 |- ( A e. V -> ( F ` A ) = [_ A / x ]_ { y | ph } )
6		csbab 4008	. . . . 5 |- [_ A / x ]_ { y | ph } = { y | [. A / x ]. ph }
7	5, 6	syl6eq 2672	. . . 4 |- ( A e. V -> ( F ` A ) = { y | [. A / x ]. ph } )
8	7	eleq2d 2687	. . 3 |- ( A e. V -> ( B e. ( F ` A ) <-> B e. { y | [. A / x ]. ph } ) )
9	8	adantr 481	. 2 |- ( ( A e. V /\ B e. W ) -> ( B e. ( F ` A ) <-> B e. { y | [. A / x ]. ph } ) )
10		simpl 473	. . . . . . 7 |- ( ( A e. V /\ y = B ) -> A e. V )
11		abfmpel.3	. . . . . . . . 9 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
12	11	ancoms 469	. . . . . . . 8 |- ( ( y = B /\ x = A ) -> ( ph <-> ps ) )
13	12	adantll 750	. . . . . . 7 |- ( ( ( A e. V /\ y = B ) /\ x = A ) -> ( ph <-> ps ) )
14	10, 13	sbcied 3472	. . . . . 6 |- ( ( A e. V /\ y = B ) -> ( [. A / x ]. ph <-> ps ) )
15	14	ex 450	. . . . 5 |- ( A e. V -> ( y = B -> ( [. A / x ]. ph <-> ps ) ) )
16	15	alrimiv 1855	. . . 4 |- ( A e. V -> A. y ( y = B -> ( [. A / x ]. ph <-> ps ) ) )
17		elabgt 3347	. . . 4 |- ( ( B e. W /\ A. y ( y = B -> ( [. A / x ]. ph <-> ps ) ) ) -> ( B e. { y | [. A / x ]. ph } <-> ps ) )
18	16, 17	sylan2 491	. . 3 |- ( ( B e. W /\ A e. V ) -> ( B e. { y | [. A / x ]. ph } <-> ps ) )
19	18	ancoms 469	. 2 |- ( ( A e. V /\ B e. W ) -> ( B e. { y | [. A / x ]. ph } <-> ps ) )
20	9, 19	bitrd 268	1 |- ( ( A e. V /\ B e. W ) -> ( B e. ( F ` A ) <-> ps ) )

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:  issiga  30174  ismeas  30262