Theorem prter1 34164

Description: Every partition generates an equivalence relation. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypothesis

Ref	Expression
prtlem18.1	|- .~ = { <. x , y >. | E. u e. A ( x e. u /\ y e. u ) }

Assertion

Ref	Expression
prter1	|- ( Prt A -> .~ Er U. A )

Distinct variable group:   x, u, y, A

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

Proof of Theorem prter1

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

Step	Hyp	Ref	Expression
1		prtlem18.1	. . . 4 |- .~ = { <. x , y >. | E. u e. A ( x e. u /\ y e. u ) }
2	1	relopabi 5245	. . 3 |- Rel .~
3	2	a1i 11	. 2 |- ( Prt A -> Rel .~ )
4	1	prtlem16 34154	. . 3 |- dom .~ = U. A
5	4	a1i 11	. 2 |- ( Prt A -> dom .~ = U. A )
6		prtlem15 34160	. . . . . 6 |- ( Prt A -> ( E. v e. A E. q e. A ( ( z e. v /\ w e. v ) /\ ( w e. q /\ p e. q ) ) -> E. r e. A ( z e. r /\ p e. r ) ) )
7	1	prtlem13 34153	. . . . . . . 8 |- ( z .~ w <-> E. v e. A ( z e. v /\ w e. v ) )
8	1	prtlem13 34153	. . . . . . . 8 |- ( w .~ p <-> E. q e. A ( w e. q /\ p e. q ) )
9	7, 8	anbi12i 733	. . . . . . 7 |- ( ( z .~ w /\ w .~ p ) <-> ( E. v e. A ( z e. v /\ w e. v ) /\ E. q e. A ( w e. q /\ p e. q ) ) )
10		reeanv 3107	. . . . . . 7 |- ( E. v e. A E. q e. A ( ( z e. v /\ w e. v ) /\ ( w e. q /\ p e. q ) ) <-> ( E. v e. A ( z e. v /\ w e. v ) /\ E. q e. A ( w e. q /\ p e. q ) ) )
11	9, 10	bitr4i 267	. . . . . 6 |- ( ( z .~ w /\ w .~ p ) <-> E. v e. A E. q e. A ( ( z e. v /\ w e. v ) /\ ( w e. q /\ p e. q ) ) )
12	1	prtlem13 34153	. . . . . 6 |- ( z .~ p <-> E. r e. A ( z e. r /\ p e. r ) )
13	6, 11, 12	3imtr4g 285	. . . . 5 |- ( Prt A -> ( ( z .~ w /\ w .~ p ) -> z .~ p ) )
14		pm3.22 465	. . . . . . 7 |- ( ( z e. v /\ w e. v ) -> ( w e. v /\ z e. v ) )
15	14	reximi 3011	. . . . . 6 |- ( E. v e. A ( z e. v /\ w e. v ) -> E. v e. A ( w e. v /\ z e. v ) )
16	1	prtlem13 34153	. . . . . 6 |- ( w .~ z <-> E. v e. A ( w e. v /\ z e. v ) )
17	15, 7, 16	3imtr4i 281	. . . . 5 |- ( z .~ w -> w .~ z )
18	13, 17	jctil 560	. . . 4 |- ( Prt A -> ( ( z .~ w -> w .~ z ) /\ ( ( z .~ w /\ w .~ p ) -> z .~ p ) ) )
19	18	alrimivv 1856	. . 3 |- ( Prt A -> A. w A. p ( ( z .~ w -> w .~ z ) /\ ( ( z .~ w /\ w .~ p ) -> z .~ p ) ) )
20	19	alrimiv 1855	. 2 |- ( Prt A -> A. z A. w A. p ( ( z .~ w -> w .~ z ) /\ ( ( z .~ w /\ w .~ p ) -> z .~ p ) ) )
21		dfer2 7743	. 2 |- ( .~ Er U. A <-> ( Rel .~ /\ dom .~ = U. A /\ A. z A. w A. p ( ( z .~ w -> w .~ z ) /\ ( ( z .~ w /\ w .~ p ) -> z .~ p ) ) ) )
22	3, 5, 20, 21	syl3anbrc 1246	1 |- ( Prt A -> .~ Er U. A )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   E.wrex 2913   U.cuni 4436   class class class wbr 4653   {copab 4712   dom cdm 5114   Rel wrel 5119   Er wer 7739   Prt wprt 34156

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-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-er 7742  df-prt 34157

This theorem is referenced by:  prtex  34165