Theorem cnvoprab 28383

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 1944	. . . . . 6 |- ( E. a E. z ( w = <. a , z >. /\ ps ) <-> E. z E. a ( w = <. a , z >. /\ ps ) )
2		nfv 1769	. . . . . . . . . . 11 |- F/ x w = <. a , z >.
3		cnvoprab.x	. . . . . . . . . . 11 |- F/ x ps
4	2, 3	nfan 2031	. . . . . . . . . 10 |- F/ x ( w = <. a , z >. /\ ps )
5	4	nfex 2050	. . . . . . . . 9 |- F/ x E. a ( w = <. a , z >. /\ ps )
6		nfv 1769	. . . . . . . . . . . 12 |- F/ y w = <. a , z >.
7		cnvoprab.y	. . . . . . . . . . . 12 |- F/ y ps
8	6, 7	nfan 2031	. . . . . . . . . . 11 |- F/ y ( w = <. a , z >. /\ ps )
9	8	nfex 2050	. . . . . . . . . 10 |- F/ y E. a ( w = <. a , z >. /\ ps )
10		opex 4664	. . . . . . . . . . 11 |- <. x , y >. e. _V
11		opeq1 4158	. . . . . . . . . . . . 13 |- ( a = <. x , y >. -> <. a , z >. = <. <. x , y >. , z >. )
12	11	eqeq2d 2481	. . . . . . . . . . . 12 |- ( a = <. x , y >. -> ( w = <. a , z >. <-> w = <. <. x , y >. , z >. ) )
13		cnvoprab.1	. . . . . . . . . . . 12 |- ( a = <. x , y >. -> ( ps <-> ph ) )
14	12, 13	anbi12d 725	. . . . . . . . . . 11 |- ( a = <. x , y >. -> ( ( w = <. a , z >. /\ ps ) <-> ( w = <. <. x , y >. , z >. /\ ph ) ) )
15	10, 14	spcev 3127	. . . . . . . . . 10 |- ( ( w = <. <. x , y >. , z >. /\ ph ) -> E. a ( w = <. a , z >. /\ ps ) )
16	9, 15	exlimi 2015	. . . . . . . . 9 |- ( E. y ( w = <. <. x , y >. , z >. /\ ph ) -> E. a ( w = <. a , z >. /\ ps ) )
17	5, 16	exlimi 2015	. . . . . . . 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 473	. . . . . . . . . 10 |- ( ( w = <. a , z >. /\ ps ) -> a e. ( _V X. _V ) )
20		fvex 5889	. . . . . . . . . . 11 |- ( 1st ` a ) e. _V
21		fvex 5889	. . . . . . . . . . 11 |- ( 2nd ` a ) e. _V
22		eqcom 2478	. . . . . . . . . . . . . . 15 |- ( ( 1st ` a ) = x <-> x = ( 1st ` a ) )
23		eqcom 2478	. . . . . . . . . . . . . . 15 |- ( ( 2nd ` a ) = y <-> y = ( 2nd ` a ) )
24	22, 23	anbi12i 711	. . . . . . . . . . . . . 14 |- ( ( ( 1st ` a ) = x /\ ( 2nd ` a ) = y ) <-> ( x = ( 1st ` a ) /\ y = ( 2nd ` a ) ) )
25		eqopi 6846	. . . . . . . . . . . . . 14 |- ( ( a e. ( _V X. _V ) /\ ( ( 1st ` a ) = x /\ ( 2nd ` a ) = y ) ) -> a = <. x , y >. )
26	24, 25	sylan2br 484	. . . . . . . . . . . . 13 |- ( ( a e. ( _V X. _V ) /\ ( x = ( 1st ` a ) /\ y = ( 2nd ` a ) ) ) -> a = <. x , y >. )
27	14	bicomd 206	. . . . . . . . . . . . 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 28187	. . . . . . . . . . 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 699	. . . . . . . . . 10 |- ( a e. ( _V X. _V ) -> ( ( w = <. a , z >. /\ ps ) -> E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) ) )
31	19, 30	mpcom 36	. . . . . . . . 9 |- ( ( w = <. a , z >. /\ ps ) -> E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) )
32	31	exlimiv 1784	. . . . . . . 8 |- ( E. a ( w = <. a , z >. /\ ps ) -> E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) )
33	17, 32	impbii 192	. . . . . . 7 |- ( E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) <-> E. a ( w = <. a , z >. /\ ps ) )
34	33	exbii 1726	. . . . . 6 |- ( E. z E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) <-> E. z E. a ( w = <. a , z >. /\ ps ) )
35		exrot3 1948	. . . . . 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 282	. . . . 5 |- ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) <-> E. a E. z ( w = <. a , z >. /\ ps ) )
37	36	abbii 2587	. . . 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 6312	. . . 4 |- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }
39		df-opab 4455	. . . 4 |- { <. a , z >. | ps } = { w | E. a E. z ( w = <. a , z >. /\ ps ) }
40	37, 38, 39	3eqtr4ri 2504	. . 3 |- { <. a , z >. | ps } = { <. <. x , y >. , z >. | ph }
41	40	cnveqi 5014	. 2 |- `' { <. a , z >. | ps } = `' { <. <. x , y >. , z >. | ph }
42		cnvopab 5243	. 2 |- `' { <. a , z >. | ps } = { <. z , a >. | ps }
43	41, 42	eqtr3i 2495	1 |- `' { <. <. x , y >. , z >. | ph } = { <. z , a >. | ps }

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   E.wex 1671   F/wnf 1675   e. wcel 1904   {cab 2457   _Vcvv 3031   <.cop 3965   {copab 4453   X. cxp 4837   `'ccnv 4838   ` cfv 5589   {coprab 6309   1stc1st 6810   2ndc2nd 6811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fv 5597  df-oprab 6312  df-1st 6812  df-2nd 6813

This theorem is referenced by:  f1od2  28384