Theorem prtlem16 34154

Description: Lemma for prtex 34165, prter2 34166 and prter3 34167. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypothesis

Ref	Expression
prtlem13.1	|- .~ = { <. x , y >. | E. u e. A ( x e. u /\ y e. u ) }

Assertion

Ref	Expression
prtlem16	|- dom .~ = U. A

Distinct variable group:   x, u, y, A

Allowed substitution hints:   .~ ( x, y, u)

Proof of Theorem prtlem16

Dummy variables v w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . 4 |- z e. _V
2	1	eldm 5321	. . 3 |- ( z e. dom .~ <-> E. w z .~ w )
3		prtlem13.1	. . . . 5 |- .~ = { <. x , y >. | E. u e. A ( x e. u /\ y e. u ) }
4	3	prtlem13 34153	. . . 4 |- ( z .~ w <-> E. v e. A ( z e. v /\ w e. v ) )
5	4	exbii 1774	. . 3 |- ( E. w z .~ w <-> E. w E. v e. A ( z e. v /\ w e. v ) )
6		elunii 4441	. . . . . . . 8 |- ( ( z e. v /\ v e. A ) -> z e. U. A )
7	6	ancoms 469	. . . . . . 7 |- ( ( v e. A /\ z e. v ) -> z e. U. A )
8	7	adantrr 753	. . . . . 6 |- ( ( v e. A /\ ( z e. v /\ w e. v ) ) -> z e. U. A )
9	8	rexlimiva 3028	. . . . 5 |- ( E. v e. A ( z e. v /\ w e. v ) -> z e. U. A )
10	9	exlimiv 1858	. . . 4 |- ( E. w E. v e. A ( z e. v /\ w e. v ) -> z e. U. A )
11		eluni2 4440	. . . . 5 |- ( z e. U. A <-> E. v e. A z e. v )
12		eleq1 2689	. . . . . . . . 9 |- ( w = z -> ( w e. v <-> z e. v ) )
13	12	anbi2d 740	. . . . . . . 8 |- ( w = z -> ( ( z e. v /\ w e. v ) <-> ( z e. v /\ z e. v ) ) )
14		pm4.24 675	. . . . . . . 8 |- ( z e. v <-> ( z e. v /\ z e. v ) )
15	13, 14	syl6bbr 278	. . . . . . 7 |- ( w = z -> ( ( z e. v /\ w e. v ) <-> z e. v ) )
16	15	rexbidv 3052	. . . . . 6 |- ( w = z -> ( E. v e. A ( z e. v /\ w e. v ) <-> E. v e. A z e. v ) )
17	1, 16	spcev 3300	. . . . 5 |- ( E. v e. A z e. v -> E. w E. v e. A ( z e. v /\ w e. v ) )
18	11, 17	sylbi 207	. . . 4 |- ( z e. U. A -> E. w E. v e. A ( z e. v /\ w e. v ) )
19	10, 18	impbii 199	. . 3 |- ( E. w E. v e. A ( z e. v /\ w e. v ) <-> z e. U. A )
20	2, 5, 19	3bitri 286	. 2 |- ( z e. dom .~ <-> z e. U. A )
21	20	eqriv 2619	1 |- dom .~ = U. A

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   U.cuni 4436   class class class wbr 4653   {copab 4712   dom cdm 5114

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-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-dm 5124

This theorem is referenced by:  prtlem400  34155  prter1  34164