Theorem cnvoprab 6301

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 1678	. . . . . 6 |- ( E. a E. z ( w = <. a , z >. /\ ps ) <-> E. z E. a ( w = <. a , z >. /\ ps ) )
2		nfv 1542	. . . . . . . . . . 11 |- F/ x w = <. a , z >.
3		cnvoprab.x	. . . . . . . . . . 11 |- F/ x ps
4	2, 3	nfan 1579	. . . . . . . . . 10 |- F/ x ( w = <. a , z >. /\ ps )
5	4	nfex 1651	. . . . . . . . 9 |- F/ x E. a ( w = <. a , z >. /\ ps )
6		nfv 1542	. . . . . . . . . . . 12 |- F/ y w = <. a , z >.
7		cnvoprab.y	. . . . . . . . . . . 12 |- F/ y ps
8	6, 7	nfan 1579	. . . . . . . . . . 11 |- F/ y ( w = <. a , z >. /\ ps )
9	8	nfex 1651	. . . . . . . . . 10 |- F/ y E. a ( w = <. a , z >. /\ ps )
10		vex 2766	. . . . . . . . . . . 12 |- x e. _V
11		vex 2766	. . . . . . . . . . . 12 |- y e. _V
12	10, 11	opex 4263	. . . . . . . . . . 11 |- <. x , y >. e. _V
13		opeq1 3809	. . . . . . . . . . . . 13 |- ( a = <. x , y >. -> <. a , z >. = <. <. x , y >. , z >. )
14	13	eqeq2d 2208	. . . . . . . . . . . 12 |- ( a = <. x , y >. -> ( w = <. a , z >. <-> w = <. <. x , y >. , z >. ) )
15		cnvoprab.1	. . . . . . . . . . . 12 |- ( a = <. x , y >. -> ( ps <-> ph ) )
16	14, 15	anbi12d 473	. . . . . . . . . . 11 |- ( a = <. x , y >. -> ( ( w = <. a , z >. /\ ps ) <-> ( w = <. <. x , y >. , z >. /\ ph ) ) )
17	12, 16	spcev 2859	. . . . . . . . . 10 |- ( ( w = <. <. x , y >. , z >. /\ ph ) -> E. a ( w = <. a , z >. /\ ps ) )
18	9, 17	exlimi 1608	. . . . . . . . 9 |- ( E. y ( w = <. <. x , y >. , z >. /\ ph ) -> E. a ( w = <. a , z >. /\ ps ) )
19	5, 18	exlimi 1608	. . . . . . . 8 |- ( E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) -> E. a ( w = <. a , z >. /\ ps ) )
20		cnvoprab.2	. . . . . . . . . . 11 |- ( ps -> a e. ( _V X. _V ) )
21	20	adantl 277	. . . . . . . . . 10 |- ( ( w = <. a , z >. /\ ps ) -> a e. ( _V X. _V ) )
22		vex 2766	. . . . . . . . . . . 12 |- a e. _V
23		1stexg 6234	. . . . . . . . . . . 12 |- ( a e. _V -> ( 1st ` a ) e. _V )
24	22, 23	ax-mp 5	. . . . . . . . . . 11 |- ( 1st ` a ) e. _V
25		2ndexg 6235	. . . . . . . . . . . 12 |- ( a e. _V -> ( 2nd ` a ) e. _V )
26	22, 25	ax-mp 5	. . . . . . . . . . 11 |- ( 2nd ` a ) e. _V
27		eqcom 2198	. . . . . . . . . . . . . . 15 |- ( ( 1st ` a ) = x <-> x = ( 1st ` a ) )
28		eqcom 2198	. . . . . . . . . . . . . . 15 |- ( ( 2nd ` a ) = y <-> y = ( 2nd ` a ) )
29	27, 28	anbi12i 460	. . . . . . . . . . . . . 14 |- ( ( ( 1st ` a ) = x /\ ( 2nd ` a ) = y ) <-> ( x = ( 1st ` a ) /\ y = ( 2nd ` a ) ) )
30		eqopi 6239	. . . . . . . . . . . . . 14 |- ( ( a e. ( _V X. _V ) /\ ( ( 1st ` a ) = x /\ ( 2nd ` a ) = y ) ) -> a = <. x , y >. )
31	29, 30	sylan2br 288	. . . . . . . . . . . . 13 |- ( ( a e. ( _V X. _V ) /\ ( x = ( 1st ` a ) /\ y = ( 2nd ` a ) ) ) -> a = <. x , y >. )
32	16	bicomd 141	. . . . . . . . . . . . 13 |- ( a = <. x , y >. -> ( ( w = <. <. x , y >. , z >. /\ ph ) <-> ( w = <. a , z >. /\ ps ) ) )
33	31, 32	syl 14	. . . . . . . . . . . 12 |- ( ( a e. ( _V X. _V ) /\ ( x = ( 1st ` a ) /\ y = ( 2nd ` a ) ) ) -> ( ( w = <. <. x , y >. , z >. /\ ph ) <-> ( w = <. a , z >. /\ ps ) ) )
34	4, 8, 33	spc2ed 6300	. . . . . . . . . . 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 ) ) )
35	24, 26, 34	mpanr12 439	. . . . . . . . . 10 |- ( a e. ( _V X. _V ) -> ( ( w = <. a , z >. /\ ps ) -> E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) ) )
36	21, 35	mpcom 36	. . . . . . . . 9 |- ( ( w = <. a , z >. /\ ps ) -> E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) )
37	36	exlimiv 1612	. . . . . . . 8 |- ( E. a ( w = <. a , z >. /\ ps ) -> E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) )
38	19, 37	impbii 126	. . . . . . 7 |- ( E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) <-> E. a ( w = <. a , z >. /\ ps ) )
39	38	exbii 1619	. . . . . 6 |- ( E. z E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) <-> E. z E. a ( w = <. a , z >. /\ ps ) )
40		exrot3 1704	. . . . . 6 |- ( E. z E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) <-> E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) )
41	1, 39, 40	3bitr2ri 209	. . . . 5 |- ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) <-> E. a E. z ( w = <. a , z >. /\ ps ) )
42	41	abbii 2312	. . . 4 |- { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) } = { w | E. a E. z ( w = <. a , z >. /\ ps ) }
43		df-oprab 5929	. . . 4 |- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }
44		df-opab 4096	. . . 4 |- { <. a , z >. | ps } = { w | E. a E. z ( w = <. a , z >. /\ ps ) }
45	42, 43, 44	3eqtr4ri 2228	. . 3 |- { <. a , z >. | ps } = { <. <. x , y >. , z >. | ph }
46	45	cnveqi 4842	. 2 |- `' { <. a , z >. | ps } = `' { <. <. x , y >. , z >. | ph }
47		cnvopab 5072	. 2 |- `' { <. a , z >. | ps } = { <. z , a >. | ps }
48	46, 47	eqtr3i 2219	1 |- `' { <. <. x , y >. , z >. | ph } = { <. z , a >. | ps }

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   F/wnf 1474   E.wex 1506   e. wcel 2167   {cab 2182   _Vcvv 2763   <.cop 3626   {copab 4094   X. cxp 4662   `'ccnv 4663   ` cfv 5259   {coprab 5926   1stc1st 6205   2ndc2nd 6206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fo 5265  df-fv 5267  df-oprab 5929  df-1st 6207  df-2nd 6208

This theorem is referenced by:  f1od2  6302