Theorem cnvoprab 6343

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 1688	. . . . . 6 |- ( E. a E. z ( w = <. a , z >. /\ ps ) <-> E. z E. a ( w = <. a , z >. /\ ps ) )
2		nfv 1552	. . . . . . . . . . 11 |- F/ x w = <. a , z >.
3		cnvoprab.x	. . . . . . . . . . 11 |- F/ x ps
4	2, 3	nfan 1589	. . . . . . . . . 10 |- F/ x ( w = <. a , z >. /\ ps )
5	4	nfex 1661	. . . . . . . . 9 |- F/ x E. a ( w = <. a , z >. /\ ps )
6		nfv 1552	. . . . . . . . . . . 12 |- F/ y w = <. a , z >.
7		cnvoprab.y	. . . . . . . . . . . 12 |- F/ y ps
8	6, 7	nfan 1589	. . . . . . . . . . 11 |- F/ y ( w = <. a , z >. /\ ps )
9	8	nfex 1661	. . . . . . . . . 10 |- F/ y E. a ( w = <. a , z >. /\ ps )
10		vex 2779	. . . . . . . . . . . 12 |- x e. _V
11		vex 2779	. . . . . . . . . . . 12 |- y e. _V
12	10, 11	opex 4291	. . . . . . . . . . 11 |- <. x , y >. e. _V
13		opeq1 3833	. . . . . . . . . . . . 13 |- ( a = <. x , y >. -> <. a , z >. = <. <. x , y >. , z >. )
14	13	eqeq2d 2219	. . . . . . . . . . . 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 2875	. . . . . . . . . 10 |- ( ( w = <. <. x , y >. , z >. /\ ph ) -> E. a ( w = <. a , z >. /\ ps ) )
18	9, 17	exlimi 1618	. . . . . . . . 9 |- ( E. y ( w = <. <. x , y >. , z >. /\ ph ) -> E. a ( w = <. a , z >. /\ ps ) )
19	5, 18	exlimi 1618	. . . . . . . 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 2779	. . . . . . . . . . . 12 |- a e. _V
23		1stexg 6276	. . . . . . . . . . . 12 |- ( a e. _V -> ( 1st ` a ) e. _V )
24	22, 23	ax-mp 5	. . . . . . . . . . 11 |- ( 1st ` a ) e. _V
25		2ndexg 6277	. . . . . . . . . . . 12 |- ( a e. _V -> ( 2nd ` a ) e. _V )
26	22, 25	ax-mp 5	. . . . . . . . . . 11 |- ( 2nd ` a ) e. _V
27		eqcom 2209	. . . . . . . . . . . . . . 15 |- ( ( 1st ` a ) = x <-> x = ( 1st ` a ) )
28		eqcom 2209	. . . . . . . . . . . . . . 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 6281	. . . . . . . . . . . . . 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 6342	. . . . . . . . . . 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 1622	. . . . . . . 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 1629	. . . . . 6 |- ( E. z E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) <-> E. z E. a ( w = <. a , z >. /\ ps ) )
40		exrot3 1714	. . . . . 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 2323	. . . 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 5971	. . . 4 |- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }
44		df-opab 4122	. . . 4 |- { <. a , z >. | ps } = { w | E. a E. z ( w = <. a , z >. /\ ps ) }
45	42, 43, 44	3eqtr4ri 2239	. . 3 |- { <. a , z >. | ps } = { <. <. x , y >. , z >. | ph }
46	45	cnveqi 4871	. 2 |- `' { <. a , z >. | ps } = `' { <. <. x , y >. , z >. | ph }
47		cnvopab 5103	. 2 |- `' { <. a , z >. | ps } = { <. z , a >. | ps }
48	46, 47	eqtr3i 2230	1 |- `' { <. <. x , y >. , z >. | ph } = { <. z , a >. | ps }

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   F/wnf 1484   E.wex 1516   e. wcel 2178   {cab 2193   _Vcvv 2776   <.cop 3646   {copab 4120   X. cxp 4691   `'ccnv 4692   ` cfv 5290   {coprab 5968   1stc1st 6247   2ndc2nd 6248

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fo 5296  df-fv 5298  df-oprab 5971  df-1st 6249  df-2nd 6250

This theorem is referenced by:  f1od2  6344