Theorem cnvoprab 26045

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 1787	. . . . . 6 |- ( E. a E. z ( w = <. a , z >. /\ ps ) <-> E. z E. a ( w = <. a , z >. /\ ps ) )
2		opex 4577	. . . . . . . . . . . . 13 |- <. x , y >. e. _V
3		opeq1 4080	. . . . . . . . . . . . . . . 16 |- ( a = <. x , y >. -> <. a , z >. = <. <. x , y >. , z >. )
4	3	eqeq2d 2454	. . . . . . . . . . . . . . 15 |- ( a = <. x , y >. -> ( w = <. a , z >. <-> w = <. <. x , y >. , z >. ) )
5		cnvoprab.1	. . . . . . . . . . . . . . 15 |- ( a = <. x , y >. -> ( ps <-> ph ) )
6	4, 5	anbi12d 710	. . . . . . . . . . . . . 14 |- ( a = <. x , y >. -> ( ( w = <. a , z >. /\ ps ) <-> ( w = <. <. x , y >. , z >. /\ ph ) ) )
7	6	spcegv 3079	. . . . . . . . . . . . 13 |- ( <. x , y >. e. _V -> ( ( w = <. <. x , y >. , z >. /\ ph ) -> E. a ( w = <. a , z >. /\ ps ) ) )
8	2, 7	ax-mp 5	. . . . . . . . . . . 12 |- ( ( w = <. <. x , y >. , z >. /\ ph ) -> E. a ( w = <. a , z >. /\ ps ) )
9	8	eximi 1625	. . . . . . . . . . 11 |- ( E. y ( w = <. <. x , y >. , z >. /\ ph ) -> E. y E. a ( w = <. a , z >. /\ ps ) )
10		nfv 1673	. . . . . . . . . . . . . 14 |- F/ y w = <. a , z >.
11		cnvoprab.y	. . . . . . . . . . . . . 14 |- F/ y ps
12	10, 11	nfan 1861	. . . . . . . . . . . . 13 |- F/ y ( w = <. a , z >. /\ ps )
13	12	nfex 1874	. . . . . . . . . . . 12 |- F/ y E. a ( w = <. a , z >. /\ ps )
14	13	19.9 1827	. . . . . . . . . . 11 |- ( E. y E. a ( w = <. a , z >. /\ ps ) <-> E. a ( w = <. a , z >. /\ ps ) )
15	9, 14	sylib 196	. . . . . . . . . 10 |- ( E. y ( w = <. <. x , y >. , z >. /\ ph ) -> E. a ( w = <. a , z >. /\ ps ) )
16	15	eximi 1625	. . . . . . . . 9 |- ( E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) -> E. x E. a ( w = <. a , z >. /\ ps ) )
17		nfv 1673	. . . . . . . . . . . 12 |- F/ x w = <. a , z >.
18		cnvoprab.x	. . . . . . . . . . . 12 |- F/ x ps
19	17, 18	nfan 1861	. . . . . . . . . . 11 |- F/ x ( w = <. a , z >. /\ ps )
20	19	nfex 1874	. . . . . . . . . 10 |- F/ x E. a ( w = <. a , z >. /\ ps )
21	20	19.9 1827	. . . . . . . . 9 |- ( E. x E. a ( w = <. a , z >. /\ ps ) <-> E. a ( w = <. a , z >. /\ ps ) )
22	16, 21	sylib 196	. . . . . . . 8 |- ( E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) -> E. a ( w = <. a , z >. /\ ps ) )
23		cnvoprab.2	. . . . . . . . . . 11 |- ( ps -> a e. ( _V X. _V ) )
24	23	adantl 466	. . . . . . . . . 10 |- ( ( w = <. a , z >. /\ ps ) -> a e. ( _V X. _V ) )
25		fvex 5722	. . . . . . . . . . 11 |- ( 1st ` a ) e. _V
26		fvex 5722	. . . . . . . . . . 11 |- ( 2nd ` a ) e. _V
27		eqcom 2445	. . . . . . . . . . . . . . . 16 |- ( ( 1st ` a ) = x <-> x = ( 1st ` a ) )
28		eqcom 2445	. . . . . . . . . . . . . . . 16 |- ( ( 2nd ` a ) = y <-> y = ( 2nd ` a ) )
29	27, 28	anbi12i 697	. . . . . . . . . . . . . . 15 |- ( ( ( 1st ` a ) = x /\ ( 2nd ` a ) = y ) <-> ( x = ( 1st ` a ) /\ y = ( 2nd ` a ) ) )
30		eqopi 6631	. . . . . . . . . . . . . . 15 |- ( ( a e. ( _V X. _V ) /\ ( ( 1st ` a ) = x /\ ( 2nd ` a ) = y ) ) -> a = <. x , y >. )
31	29, 30	sylan2br 476	. . . . . . . . . . . . . 14 |- ( ( a e. ( _V X. _V ) /\ ( x = ( 1st ` a ) /\ y = ( 2nd ` a ) ) ) -> a = <. x , y >. )
32	6	bicomd 201	. . . . . . . . . . . . . 14 |- ( a = <. x , y >. -> ( ( w = <. <. x , y >. , z >. /\ ph ) <-> ( w = <. a , z >. /\ ps ) ) )
33	31, 32	syl 16	. . . . . . . . . . . . 13 |- ( ( a e. ( _V X. _V ) /\ ( x = ( 1st ` a ) /\ y = ( 2nd ` a ) ) ) -> ( ( w = <. <. x , y >. , z >. /\ ph ) <-> ( w = <. a , z >. /\ ps ) ) )
34	19, 12, 33	spc2ed 25879	. . . . . . . . . . . 12 |- ( ( 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 ) ) )
35	34	ex 434	. . . . . . . . . . 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 ) ) ) )
36	25, 26, 35	mp2ani 678	. . . . . . . . . 10 |- ( a e. ( _V X. _V ) -> ( ( w = <. a , z >. /\ ps ) -> E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) ) )
37	24, 36	mpcom 36	. . . . . . . . 9 |- ( ( w = <. a , z >. /\ ps ) -> E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) )
38	37	exlimiv 1688	. . . . . . . 8 |- ( E. a ( w = <. a , z >. /\ ps ) -> E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) )
39	22, 38	impbii 188	. . . . . . 7 |- ( E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) <-> E. a ( w = <. a , z >. /\ ps ) )
40	39	exbii 1634	. . . . . 6 |- ( E. z E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) <-> E. z E. a ( w = <. a , z >. /\ ps ) )
41		excom 1787	. . . . . . . 8 |- ( E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) <-> E. z E. y ( w = <. <. x , y >. , z >. /\ ph ) )
42	41	exbii 1634	. . . . . . 7 |- ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) <-> E. x E. z E. y ( w = <. <. x , y >. , z >. /\ ph ) )
43		excom 1787	. . . . . . 7 |- ( E. x E. z E. y ( w = <. <. x , y >. , z >. /\ ph ) <-> E. z E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) )
44	42, 43	bitr2i 250	. . . . . 6 |- ( E. z E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) <-> E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) )
45	1, 40, 44	3bitr2ri 274	. . . . 5 |- ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) <-> E. a E. z ( w = <. a , z >. /\ ps ) )
46	45	abbii 2561	. . . 4 |- { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) } = { w | E. a E. z ( w = <. a , z >. /\ ps ) }
47		df-oprab 6116	. . . 4 |- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }
48		df-opab 4372	. . . 4 |- { <. a , z >. | ps } = { w | E. a E. z ( w = <. a , z >. /\ ps ) }
49	46, 47, 48	3eqtr4ri 2474	. . 3 |- { <. a , z >. | ps } = { <. <. x , y >. , z >. | ph }
50	49	cnveqi 5035	. 2 |- `' { <. a , z >. | ps } = `' { <. <. x , y >. , z >. | ph }
51		cnvopab 5259	. 2 |- `' { <. a , z >. | ps } = { <. z , a >. | ps }
52	50, 51	eqtr3i 2465	1 |- `' { <. <. x , y >. , z >. | ph } = { <. z , a >. | ps }

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   E.wex 1586   F/wnf 1589   e. wcel 1756   {cab 2429   _Vcvv 2993   <.cop 3904   {copab 4370   X. cxp 4859   `'ccnv 4860   ` cfv 5439   {coprab 6113   1stc1st 6596   2ndc2nd 6597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-iota 5402  df-fun 5441  df-fv 5447  df-oprab 6116  df-1st 6598  df-2nd 6599

This theorem is referenced by:  f1od2  26046