Theorem prtlem10 34150

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

Assertion

Ref	Expression
prtlem10	|- ( .~ Er A -> ( z e. A -> ( z .~ w <-> E. v e. A ( z e. [ v ] .~ /\ w e. [ v ] .~ ) ) ) )

Distinct variable groups:   w, v   z, v   v, A   v, .~

Allowed substitution hints:   A( z, w)   .~ ( z, w)

Proof of Theorem prtlem10

Step	Hyp	Ref	Expression
1		simpr 477	. . . . 5 |- ( ( .~ Er A /\ z e. A ) -> z e. A )
2		simpl 473	. . . . . 6 |- ( ( .~ Er A /\ z e. A ) -> .~ Er A )
3	2, 1	erref 7762	. . . . 5 |- ( ( .~ Er A /\ z e. A ) -> z .~ z )
4		breq1 4656	. . . . . . . 8 |- ( v = z -> ( v .~ z <-> z .~ z ) )
5		breq1 4656	. . . . . . . 8 |- ( v = z -> ( v .~ w <-> z .~ w ) )
6	4, 5	anbi12d 747	. . . . . . 7 |- ( v = z -> ( ( v .~ z /\ v .~ w ) <-> ( z .~ z /\ z .~ w ) ) )
7	6	rspcev 3309	. . . . . 6 |- ( ( z e. A /\ ( z .~ z /\ z .~ w ) ) -> E. v e. A ( v .~ z /\ v .~ w ) )
8	7	expr 643	. . . . 5 |- ( ( z e. A /\ z .~ z ) -> ( z .~ w -> E. v e. A ( v .~ z /\ v .~ w ) ) )
9	1, 3, 8	syl2anc 693	. . . 4 |- ( ( .~ Er A /\ z e. A ) -> ( z .~ w -> E. v e. A ( v .~ z /\ v .~ w ) ) )
10		simplll 798	. . . . . . 7 |- ( ( ( ( .~ Er A /\ z e. A ) /\ v e. A ) /\ ( v .~ z /\ v .~ w ) ) -> .~ Er A )
11		simprl 794	. . . . . . 7 |- ( ( ( ( .~ Er A /\ z e. A ) /\ v e. A ) /\ ( v .~ z /\ v .~ w ) ) -> v .~ z )
12		simprr 796	. . . . . . 7 |- ( ( ( ( .~ Er A /\ z e. A ) /\ v e. A ) /\ ( v .~ z /\ v .~ w ) ) -> v .~ w )
13	10, 11, 12	ertr3d 7760	. . . . . 6 |- ( ( ( ( .~ Er A /\ z e. A ) /\ v e. A ) /\ ( v .~ z /\ v .~ w ) ) -> z .~ w )
14	13	ex 450	. . . . 5 |- ( ( ( .~ Er A /\ z e. A ) /\ v e. A ) -> ( ( v .~ z /\ v .~ w ) -> z .~ w ) )
15	14	rexlimdva 3031	. . . 4 |- ( ( .~ Er A /\ z e. A ) -> ( E. v e. A ( v .~ z /\ v .~ w ) -> z .~ w ) )
16	9, 15	impbid 202	. . 3 |- ( ( .~ Er A /\ z e. A ) -> ( z .~ w <-> E. v e. A ( v .~ z /\ v .~ w ) ) )
17		vex 3203	. . . . . 6 |- z e. _V
18		vex 3203	. . . . . 6 |- v e. _V
19	17, 18	elec 7786	. . . . 5 |- ( z e. [ v ] .~ <-> v .~ z )
20		vex 3203	. . . . . 6 |- w e. _V
21	20, 18	elec 7786	. . . . 5 |- ( w e. [ v ] .~ <-> v .~ w )
22	19, 21	anbi12i 733	. . . 4 |- ( ( z e. [ v ] .~ /\ w e. [ v ] .~ ) <-> ( v .~ z /\ v .~ w ) )
23	22	rexbii 3041	. . 3 |- ( E. v e. A ( z e. [ v ] .~ /\ w e. [ v ] .~ ) <-> E. v e. A ( v .~ z /\ v .~ w ) )
24	16, 23	syl6bbr 278	. 2 |- ( ( .~ Er A /\ z e. A ) -> ( z .~ w <-> E. v e. A ( z e. [ v ] .~ /\ w e. [ v ] .~ ) ) )
25	24	ex 450	1 |- ( .~ Er A -> ( z e. A -> ( z .~ w <-> E. v e. A ( z e. [ v ] .~ /\ w e. [ v ] .~ ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   E.wrex 2913   class class class wbr 4653   Er wer 7739   [cec 7740

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-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-er 7742  df-ec 7744

This theorem is referenced by: (None)