Theorem cnvoprab 28159

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 1901	. . . . . 6 |- ( E. a E. z ( w = <. a , z >. /\ ps ) <-> E. z E. a ( w = <. a , z >. /\ ps ) )
2		nfv 1754	. . . . . . . . . . 11 |- F/ x w = <. a , z >.
3		cnvoprab.x	. . . . . . . . . . 11 |- F/ x ps
4	2, 3	nfan 1986	. . . . . . . . . 10 |- F/ x ( w = <. a , z >. /\ ps )
5	4	nfex 2006	. . . . . . . . 9 |- F/ x E. a ( w = <. a , z >. /\ ps )
6		nfv 1754	. . . . . . . . . . . 12 |- F/ y w = <. a , z >.
7		cnvoprab.y	. . . . . . . . . . . 12 |- F/ y ps
8	6, 7	nfan 1986	. . . . . . . . . . 11 |- F/ y ( w = <. a , z >. /\ ps )
9	8	nfex 2006	. . . . . . . . . 10 |- F/ y E. a ( w = <. a , z >. /\ ps )
10		opex 4686	. . . . . . . . . . 11 |- <. x , y >. e. _V
11		opeq1 4190	. . . . . . . . . . . . 13 |- ( a = <. x , y >. -> <. a , z >. = <. <. x , y >. , z >. )
12	11	eqeq2d 2443	. . . . . . . . . . . 12 |- ( a = <. x , y >. -> ( w = <. a , z >. <-> w = <. <. x , y >. , z >. ) )
13		cnvoprab.1	. . . . . . . . . . . 12 |- ( a = <. x , y >. -> ( ps <-> ph ) )
14	12, 13	anbi12d 715	. . . . . . . . . . 11 |- ( a = <. x , y >. -> ( ( w = <. a , z >. /\ ps ) <-> ( w = <. <. x , y >. , z >. /\ ph ) ) )
15	10, 14	spcev 3179	. . . . . . . . . 10 |- ( ( w = <. <. x , y >. , z >. /\ ph ) -> E. a ( w = <. a , z >. /\ ps ) )
16	9, 15	exlimi 1970	. . . . . . . . 9 |- ( E. y ( w = <. <. x , y >. , z >. /\ ph ) -> E. a ( w = <. a , z >. /\ ps ) )
17	5, 16	exlimi 1970	. . . . . . . 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 467	. . . . . . . . . 10 |- ( ( w = <. a , z >. /\ ps ) -> a e. ( _V X. _V ) )
20		fvex 5891	. . . . . . . . . . 11 |- ( 1st ` a ) e. _V
21		fvex 5891	. . . . . . . . . . 11 |- ( 2nd ` a ) e. _V
22		eqcom 2438	. . . . . . . . . . . . . . 15 |- ( ( 1st ` a ) = x <-> x = ( 1st ` a ) )
23		eqcom 2438	. . . . . . . . . . . . . . 15 |- ( ( 2nd ` a ) = y <-> y = ( 2nd ` a ) )
24	22, 23	anbi12i 701	. . . . . . . . . . . . . 14 |- ( ( ( 1st ` a ) = x /\ ( 2nd ` a ) = y ) <-> ( x = ( 1st ` a ) /\ y = ( 2nd ` a ) ) )
25		eqopi 6841	. . . . . . . . . . . . . 14 |- ( ( a e. ( _V X. _V ) /\ ( ( 1st ` a ) = x /\ ( 2nd ` a ) = y ) ) -> a = <. x , y >. )
26	24, 25	sylan2br 478	. . . . . . . . . . . . 13 |- ( ( a e. ( _V X. _V ) /\ ( x = ( 1st ` a ) /\ y = ( 2nd ` a ) ) ) -> a = <. x , y >. )
27	14	bicomd 204	. . . . . . . . . . . . 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 27949	. . . . . . . . . . 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 689	. . . . . . . . . 10 |- ( a e. ( _V X. _V ) -> ( ( w = <. a , z >. /\ ps ) -> E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) ) )
31	19, 30	mpcom 37	. . . . . . . . 9 |- ( ( w = <. a , z >. /\ ps ) -> E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) )
32	31	exlimiv 1769	. . . . . . . 8 |- ( E. a ( w = <. a , z >. /\ ps ) -> E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) )
33	17, 32	impbii 190	. . . . . . 7 |- ( E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) <-> E. a ( w = <. a , z >. /\ ps ) )
34	33	exbii 1714	. . . . . 6 |- ( E. z E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) <-> E. z E. a ( w = <. a , z >. /\ ps ) )
35		exrot3 1905	. . . . . 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 277	. . . . 5 |- ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) <-> E. a E. z ( w = <. a , z >. /\ ps ) )
37	36	abbii 2563	. . . 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 6309	. . . 4 |- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }
39		df-opab 4485	. . . 4 |- { <. a , z >. | ps } = { w | E. a E. z ( w = <. a , z >. /\ ps ) }
40	37, 38, 39	3eqtr4ri 2469	. . 3 |- { <. a , z >. | ps } = { <. <. x , y >. , z >. | ph }
41	40	cnveqi 5029	. 2 |- `' { <. a , z >. | ps } = `' { <. <. x , y >. , z >. | ph }
42		cnvopab 5257	. 2 |- `' { <. a , z >. | ps } = { <. z , a >. | ps }
43	41, 42	eqtr3i 2460	1 |- `' { <. <. x , y >. , z >. | ph } = { <. z , a >. | ps }

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   E.wex 1659   F/wnf 1663   e. wcel 1870   {cab 2414   _Vcvv 3087   <.cop 4008   {copab 4483   X. cxp 4852   `'ccnv 4853   ` cfv 5601   {coprab 6306   1stc1st 6805   2ndc2nd 6806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fv 5609  df-oprab 6309  df-1st 6807  df-2nd 6808

This theorem is referenced by:  f1od2  28160