Theorem cnvoprab 29498

Description: The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.)

Hypotheses

Ref	Expression
cnvoprab.x	|- F/ x ps
cnvoprab.y	|- F/ y ps
cnvoprab.1	|- ( a = <. x , y >. -> ( ps <-> ph ) )
cnvoprab.2	|- ( ps -> a e. ( _V X. _V ) )

Assertion

Ref	Expression
cnvoprab	|- `' { <. <. x , y >. , z >. | ph } = { <. z , a >. | ps }

Distinct variable groups:   x, a, y, z   ph, a

Allowed substitution hints:   ph( x, y, z)   ps( x, y, z, a)

Proof of Theorem cnvoprab

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		excom 2042	. . . . . 6 |- ( E. a E. z ( w = <. a , z >. /\ ps ) <-> E. z E. a ( w = <. a , z >. /\ ps ) )
2		nfv 1843	. . . . . . . . . . 11 |- F/ x w = <. a , z >.
3		cnvoprab.x	. . . . . . . . . . 11 |- F/ x ps
4	2, 3	nfan 1828	. . . . . . . . . 10 |- F/ x ( w = <. a , z >. /\ ps )
5	4	nfex 2154	. . . . . . . . 9 |- F/ x E. a ( w = <. a , z >. /\ ps )
6		nfv 1843	. . . . . . . . . . . 12 |- F/ y w = <. a , z >.
7		cnvoprab.y	. . . . . . . . . . . 12 |- F/ y ps
8	6, 7	nfan 1828	. . . . . . . . . . 11 |- F/ y ( w = <. a , z >. /\ ps )
9	8	nfex 2154	. . . . . . . . . 10 |- F/ y E. a ( w = <. a , z >. /\ ps )
10		opex 4932	. . . . . . . . . . 11 |- <. x , y >. e. _V
11		opeq1 4402	. . . . . . . . . . . . 13 |- ( a = <. x , y >. -> <. a , z >. = <. <. x , y >. , z >. )
12	11	eqeq2d 2632	. . . . . . . . . . . 12 |- ( a = <. x , y >. -> ( w = <. a , z >. <-> w = <. <. x , y >. , z >. ) )
13		cnvoprab.1	. . . . . . . . . . . 12 |- ( a = <. x , y >. -> ( ps <-> ph ) )
14	12, 13	anbi12d 747	. . . . . . . . . . 11 |- ( a = <. x , y >. -> ( ( w = <. a , z >. /\ ps ) <-> ( w = <. <. x , y >. , z >. /\ ph ) ) )
15	10, 14	spcev 3300	. . . . . . . . . 10 |- ( ( w = <. <. x , y >. , z >. /\ ph ) -> E. a ( w = <. a , z >. /\ ps ) )
16	9, 15	exlimi 2086	. . . . . . . . 9 |- ( E. y ( w = <. <. x , y >. , z >. /\ ph ) -> E. a ( w = <. a , z >. /\ ps ) )
17	5, 16	exlimi 2086	. . . . . . . 8 |- ( E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) -> E. a ( w = <. a , z >. /\ ps ) )
18		cnvoprab.2	. . . . . . . . . . 11 |- ( ps -> a e. ( _V X. _V ) )
19	18	adantl 482	. . . . . . . . . 10 |- ( ( w = <. a , z >. /\ ps ) -> a e. ( _V X. _V ) )
20		fvex 6201	. . . . . . . . . . 11 |- ( 1st ` a ) e. _V
21		fvex 6201	. . . . . . . . . . 11 |- ( 2nd ` a ) e. _V
22		eqcom 2629	. . . . . . . . . . . . . . 15 |- ( ( 1st ` a ) = x <-> x = ( 1st ` a ) )
23		eqcom 2629	. . . . . . . . . . . . . . 15 |- ( ( 2nd ` a ) = y <-> y = ( 2nd ` a ) )
24	22, 23	anbi12i 733	. . . . . . . . . . . . . 14 |- ( ( ( 1st ` a ) = x /\ ( 2nd ` a ) = y ) <-> ( x = ( 1st ` a ) /\ y = ( 2nd ` a ) ) )
25		eqopi 7202	. . . . . . . . . . . . . 14 |- ( ( a e. ( _V X. _V ) /\ ( ( 1st ` a ) = x /\ ( 2nd ` a ) = y ) ) -> a = <. x , y >. )
26	24, 25	sylan2br 493	. . . . . . . . . . . . 13 |- ( ( a e. ( _V X. _V ) /\ ( x = ( 1st ` a ) /\ y = ( 2nd ` a ) ) ) -> a = <. x , y >. )
27	14	bicomd 213	. . . . . . . . . . . . 13 |- ( a = <. x , y >. -> ( ( w = <. <. x , y >. , z >. /\ ph ) <-> ( w = <. a , z >. /\ ps ) ) )
28	26, 27	syl 17	. . . . . . . . . . . 12 |- ( ( a e. ( _V X. _V ) /\ ( x = ( 1st ` a ) /\ y = ( 2nd ` a ) ) ) -> ( ( w = <. <. x , y >. , z >. /\ ph ) <-> ( w = <. a , z >. /\ ps ) ) )
29	4, 8, 28	spc2ed 29312	. . . . . . . . . . 11 |- ( ( a e. ( _V X. _V ) /\ ( ( 1st ` a ) e. _V /\ ( 2nd ` a ) e. _V ) ) -> ( ( w = <. a , z >. /\ ps ) -> E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) ) )
30	20, 21, 29	mpanr12 721	. . . . . . . . . 10 |- ( a e. ( _V X. _V ) -> ( ( w = <. a , z >. /\ ps ) -> E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) ) )
31	19, 30	mpcom 38	. . . . . . . . 9 |- ( ( w = <. a , z >. /\ ps ) -> E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) )
32	31	exlimiv 1858	. . . . . . . 8 |- ( E. a ( w = <. a , z >. /\ ps ) -> E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) )
33	17, 32	impbii 199	. . . . . . 7 |- ( E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) <-> E. a ( w = <. a , z >. /\ ps ) )
34	33	exbii 1774	. . . . . 6 |- ( E. z E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) <-> E. z E. a ( w = <. a , z >. /\ ps ) )
35		exrot3 2045	. . . . . 6 |- ( E. z E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) <-> E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) )
36	1, 34, 35	3bitr2ri 289	. . . . 5 |- ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) <-> E. a E. z ( w = <. a , z >. /\ ps ) )
37	36	abbii 2739	. . . 4 |- { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) } = { w | E. a E. z ( w = <. a , z >. /\ ps ) }
38		df-oprab 6654	. . . 4 |- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }
39		df-opab 4713	. . . 4 |- { <. a , z >. | ps } = { w | E. a E. z ( w = <. a , z >. /\ ps ) }
40	37, 38, 39	3eqtr4ri 2655	. . 3 |- { <. a , z >. | ps } = { <. <. x , y >. , z >. | ph }
41	40	cnveqi 5297	. 2 |- `' { <. a , z >. | ps } = `' { <. <. x , y >. , z >. | ph }
42		cnvopab 5533	. 2 |- `' { <. a , z >. | ps } = { <. z , a >. | ps }
43	41, 42	eqtr3i 2646	1 |- `' { <. <. x , y >. , z >. | ph } = { <. z , a >. | ps }

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   F/wnf 1708   e. wcel 1990   {cab 2608   _Vcvv 3200   <.cop 4183   {copab 4712   X. cxp 5112   `'ccnv 5113   ` cfv 5888   {coprab 6651   1stc1st 7166   2ndc2nd 7167

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-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-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-oprab 6654  df-1st 7168  df-2nd 7169

This theorem is referenced by:  f1od2  29499