Theorem eldmqsres 34051

Description: Elementhood in a restricted domain quotient set. (Contributed by Peter Mazsa, 21-Aug-2020.)

Assertion

Ref	Expression
eldmqsres	|- ( B e. V -> ( B e. ( dom ( R |` A ) /. ( R |` A ) ) <-> E. u e. A ( E. x x e. [ u ] R /\ B = [ u ] R ) ) )

Distinct variable groups:   u, A, x   u, B   u, R, x

Allowed substitution hints:   B( x)   V( x, u)

Proof of Theorem eldmqsres

Step	Hyp	Ref	Expression
1		elqsg 7798	. 2 |- ( B e. V -> ( B e. ( dom ( R |` A ) /. ( R |` A ) ) <-> E. u e. dom ( R |` A ) B = [ u ] ( R |` A ) ) )
2		eldmres2 34038	. . . . . 6 |- ( u e. _V -> ( u e. dom ( R |` A ) <-> ( u e. A /\ E. x x e. [ u ] R ) ) )
3	2	elv 33983	. . . . 5 |- ( u e. dom ( R |` A ) <-> ( u e. A /\ E. x x e. [ u ] R ) )
4	3	anbi1i 731	. . . 4 |- ( ( u e. dom ( R |` A ) /\ B = [ u ] ( R |` A ) ) <-> ( ( u e. A /\ E. x x e. [ u ] R ) /\ B = [ u ] ( R |` A ) ) )
5		ecres2 34044	. . . . . . . 8 |- ( u e. A -> [ u ] ( R |` A ) = [ u ] R )
6	5	eqeq2d 2632	. . . . . . 7 |- ( u e. A -> ( B = [ u ] ( R |` A ) <-> B = [ u ] R ) )
7	6	pm5.32i 669	. . . . . 6 |- ( ( u e. A /\ B = [ u ] ( R |` A ) ) <-> ( u e. A /\ B = [ u ] R ) )
8	7	anbi2i 730	. . . . 5 |- ( ( E. x x e. [ u ] R /\ ( u e. A /\ B = [ u ] ( R |` A ) ) ) <-> ( E. x x e. [ u ] R /\ ( u e. A /\ B = [ u ] R ) ) )
9		3ancoma 1045	. . . . . 6 |- ( ( u e. A /\ E. x x e. [ u ] R /\ B = [ u ] ( R |` A ) ) <-> ( E. x x e. [ u ] R /\ u e. A /\ B = [ u ] ( R |` A ) ) )
10		df-3an 1039	. . . . . 6 |- ( ( u e. A /\ E. x x e. [ u ] R /\ B = [ u ] ( R |` A ) ) <-> ( ( u e. A /\ E. x x e. [ u ] R ) /\ B = [ u ] ( R |` A ) ) )
11		3anass 1042	. . . . . 6 |- ( ( E. x x e. [ u ] R /\ u e. A /\ B = [ u ] ( R |` A ) ) <-> ( E. x x e. [ u ] R /\ ( u e. A /\ B = [ u ] ( R |` A ) ) ) )
12	9, 10, 11	3bitr3i 290	. . . . 5 |- ( ( ( u e. A /\ E. x x e. [ u ] R ) /\ B = [ u ] ( R |` A ) ) <-> ( E. x x e. [ u ] R /\ ( u e. A /\ B = [ u ] ( R |` A ) ) ) )
13		an12 838	. . . . 5 |- ( ( u e. A /\ ( E. x x e. [ u ] R /\ B = [ u ] R ) ) <-> ( E. x x e. [ u ] R /\ ( u e. A /\ B = [ u ] R ) ) )
14	8, 12, 13	3bitr4i 292	. . . 4 |- ( ( ( u e. A /\ E. x x e. [ u ] R ) /\ B = [ u ] ( R |` A ) ) <-> ( u e. A /\ ( E. x x e. [ u ] R /\ B = [ u ] R ) ) )
15	4, 14	bitri 264	. . 3 |- ( ( u e. dom ( R |` A ) /\ B = [ u ] ( R |` A ) ) <-> ( u e. A /\ ( E. x x e. [ u ] R /\ B = [ u ] R ) ) )
16	15	rexbii2 3039	. 2 |- ( E. u e. dom ( R |` A ) B = [ u ] ( R |` A ) <-> E. u e. A ( E. x x e. [ u ] R /\ B = [ u ] R ) )
17	1, 16	syl6bb 276	1 |- ( B e. V -> ( B e. ( dom ( R |` A ) /. ( R |` A ) ) <-> E. u e. A ( E. x x e. [ u ] R /\ B = [ u ] R ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   dom cdm 5114   |` cres 5116   [cec 7740   /.cqs 7741

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-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744  df-qs 7748

This theorem is referenced by:  eldmqsres2  34052