Theorem cnvoprabOLD

Description: The converse of a class abstraction of nested ordered pairs. Obsolete version of cnvoprab as of 16-Oct-2022, which has nonfreeness hypotheses instead of disjoint variable conditions. (Contributed by Thierry Arnoux, 17-Aug-2017) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cnvoprabOLD.x	|- F/ x ps
	cnvoprabOLD.y	|- F/ y ps
	cnvoprabOLD.1	|- ( a = <. x , y >. -> ( ps <-> ph ) )
	cnvoprabOLD.2	|- ( ps -> a e. ( _V X. _V ) )
Assertion	cnvoprabOLD	|- `' { <. <. x , y >. , z >. | ph } = { <. z , a >. | ps }

Proof

Step	Hyp	Ref	Expression
1		cnvoprabOLD.x	|- F/ x ps
2		cnvoprabOLD.y	|- F/ y ps
3		cnvoprabOLD.1	|- ( a = <. x , y >. -> ( ps <-> ph ) )
4		cnvoprabOLD.2	|- ( ps -> a e. ( _V X. _V ) )
5		excom	|- ( E. a E. z ( w = <. a , z >. /\ ps ) <-> E. z E. a ( w = <. a , z >. /\ ps ) )
6		nfv	|- F/ x w = <. a , z >.
7	6 1	nfan	|- F/ x ( w = <. a , z >. /\ ps )
8	7	nfex	|- F/ x E. a ( w = <. a , z >. /\ ps )
9		nfv	|- F/ y w = <. a , z >.
10	9 2	nfan	|- F/ y ( w = <. a , z >. /\ ps )
11	10	nfex	|- F/ y E. a ( w = <. a , z >. /\ ps )
12		opex	|- <. x , y >. e. _V
13		opeq1	|- ( a = <. x , y >. -> <. a , z >. = <. <. x , y >. , z >. )
14	13	eqeq2d	|- ( a = <. x , y >. -> ( w = <. a , z >. <-> w = <. <. x , y >. , z >. ) )
15	14 3	anbi12d	|- ( a = <. x , y >. -> ( ( w = <. a , z >. /\ ps ) <-> ( w = <. <. x , y >. , z >. /\ ph ) ) )
16	12 15	spcev	|- ( ( w = <. <. x , y >. , z >. /\ ph ) -> E. a ( w = <. a , z >. /\ ps ) )
17	11 16	exlimi	|- ( E. y ( w = <. <. x , y >. , z >. /\ ph ) -> E. a ( w = <. a , z >. /\ ps ) )
18	8 17	exlimi	|- ( E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) -> E. a ( w = <. a , z >. /\ ps ) )
19	4	adantl	|- ( ( w = <. a , z >. /\ ps ) -> a e. ( _V X. _V ) )
20		fvex	|- ( 1st ` a ) e. _V
21		fvex	|- ( 2nd ` a ) e. _V
22		eqcom	|- ( ( 1st ` a ) = x <-> x = ( 1st ` a ) )
23		eqcom	|- ( ( 2nd ` a ) = y <-> y = ( 2nd ` a ) )
24	22 23	anbi12i	|- ( ( ( 1st ` a ) = x /\ ( 2nd ` a ) = y ) <-> ( x = ( 1st ` a ) /\ y = ( 2nd ` a ) ) )
25		eqopi	|- ( ( a e. ( _V X. _V ) /\ ( ( 1st ` a ) = x /\ ( 2nd ` a ) = y ) ) -> a = <. x , y >. )
26	24 25	sylan2br	|- ( ( a e. ( _V X. _V ) /\ ( x = ( 1st ` a ) /\ y = ( 2nd ` a ) ) ) -> a = <. x , y >. )
27	15	bicomd	|- ( a = <. x , y >. -> ( ( w = <. <. x , y >. , z >. /\ ph ) <-> ( w = <. a , z >. /\ ps ) ) )
28	26 27	syl	|- ( ( a e. ( _V X. _V ) /\ ( x = ( 1st ` a ) /\ y = ( 2nd ` a ) ) ) -> ( ( w = <. <. x , y >. , z >. /\ ph ) <-> ( w = <. a , z >. /\ ps ) ) )
29	7 10 28	spc2ed	|- ( ( 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	|- ( a e. ( _V X. _V ) -> ( ( w = <. a , z >. /\ ps ) -> E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) ) )
31	19 30	mpcom	|- ( ( w = <. a , z >. /\ ps ) -> E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) )
32	31	exlimiv	|- ( E. a ( w = <. a , z >. /\ ps ) -> E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) )
33	18 32	impbii	|- ( E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) <-> E. a ( w = <. a , z >. /\ ps ) )
34	33	exbii	|- ( E. z E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) <-> E. z E. a ( w = <. a , z >. /\ ps ) )
35		exrot3	|- ( E. z E. x E. y ( w = <. <. x , y >. , z >. /\ ph ) <-> E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) )
36	5 34 35	3bitr2ri	|- ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) <-> E. a E. z ( w = <. a , z >. /\ ps ) )
37	36	abbii	|- { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) } = { w | E. a E. z ( w = <. a , z >. /\ ps ) }
38		df-oprab	|- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }
39		df-opab	|- { <. a , z >. | ps } = { w | E. a E. z ( w = <. a , z >. /\ ps ) }
40	37 38 39	3eqtr4ri	|- { <. a , z >. | ps } = { <. <. x , y >. , z >. | ph }
41	40	cnveqi	|- `' { <. a , z >. | ps } = `' { <. <. x , y >. , z >. | ph }
42		cnvopab	|- `' { <. a , z >. | ps } = { <. z , a >. | ps }
43	41 42	eqtr3i	|- `' { <. <. x , y >. , z >. | ph } = { <. z , a >. | ps }