Theorem clcnvlem 36230

Description: When A, an upper bound of the closure, exists and certain substitutions hold the converse of the closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)

Hypotheses

Ref	Expression
clcnvlem.sub1	|- ( ( ph /\ x = ( `' y u. ( X \ `' `' X ) ) ) -> ( ch -> ps ) )
clcnvlem.sub2	|- ( ( ph /\ y = `' x ) -> ( ps -> ch ) )
clcnvlem.sub3	|- ( x = A -> ( ps <-> th ) )
clcnvlem.ssub	|- ( ph -> X C_ A )
clcnvlem.ubex	|- ( ph -> A e. _V )
clcnvlem.clex	|- ( ph -> th )

Assertion

Ref	Expression
clcnvlem	|- ( ph -> `' |^| { x | ( X C_ x /\ ps ) } = |^| { y | ( `' X C_ y /\ ch ) } )

Distinct variable groups:   x, A   x, y, X   ph, x, y   ps, y   ch, x   th, x

Allowed substitution hints:   ps( x)   ch( y)   th( y)   A( y)

Proof of Theorem clcnvlem

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clcnvlem.ubex	. . . 4 |- ( ph -> A e. _V )
2		clcnvlem.ssub	. . . . 5 |- ( ph -> X C_ A )
3		clcnvlem.clex	. . . . 5 |- ( ph -> th )
4	2, 3	jca 535	. . . 4 |- ( ph -> ( X C_ A /\ th ) )
5		clcnvlem.sub3	. . . . 5 |- ( x = A -> ( ps <-> th ) )
6	5	cleq2lem 36214	. . . 4 |- ( x = A -> ( ( X C_ x /\ ps ) <-> ( X C_ A /\ th ) ) )
7	1, 4, 6	elabd 36178	. . 3 |- ( ph -> E. x ( X C_ x /\ ps ) )
8	7	cnvintabd 36209	. 2 |- ( ph -> `' |^| { x | ( X C_ x /\ ps ) } = |^| { z e. ~P ( _V X. _V ) | E. x ( z = `' x /\ ( X C_ x /\ ps ) ) } )
9		df-rab 2746	. . . . 5 |- { z e. ~P ( _V X. _V ) | E. x ( z = `' x /\ ( X C_ x /\ ps ) ) } = { z | ( z e. ~P ( _V X. _V ) /\ E. x ( z = `' x /\ ( X C_ x /\ ps ) ) ) }
10		exsimpl 1729	. . . . . . . . . . 11 |- ( E. x ( z = `' x /\ ( X C_ x /\ ps ) ) -> E. x z = `' x )
11		relcnv 5207	. . . . . . . . . . . . 13 |- Rel `' x
12		releq 4917	. . . . . . . . . . . . 13 |- ( z = `' x -> ( Rel z <-> Rel `' x ) )
13	11, 12	mpbiri 237	. . . . . . . . . . . 12 |- ( z = `' x -> Rel z )
14	13	exlimiv 1776	. . . . . . . . . . 11 |- ( E. x z = `' x -> Rel z )
15	10, 14	syl 17	. . . . . . . . . 10 |- ( E. x ( z = `' x /\ ( X C_ x /\ ps ) ) -> Rel z )
16		df-rel 4841	. . . . . . . . . 10 |- ( Rel z <-> z C_ ( _V X. _V ) )
17	15, 16	sylib 200	. . . . . . . . 9 |- ( E. x ( z = `' x /\ ( X C_ x /\ ps ) ) -> z C_ ( _V X. _V ) )
18		selpw 3958	. . . . . . . . . 10 |- ( z e. ~P ( _V X. _V ) <-> z C_ ( _V X. _V ) )
19	18	bicomi 206	. . . . . . . . 9 |- ( z C_ ( _V X. _V ) <-> z e. ~P ( _V X. _V ) )
20	17, 19	sylib 200	. . . . . . . 8 |- ( E. x ( z = `' x /\ ( X C_ x /\ ps ) ) -> z e. ~P ( _V X. _V ) )
21	20	pm4.71ri 639	. . . . . . 7 |- ( E. x ( z = `' x /\ ( X C_ x /\ ps ) ) <-> ( z e. ~P ( _V X. _V ) /\ E. x ( z = `' x /\ ( X C_ x /\ ps ) ) ) )
22	21	bicomi 206	. . . . . 6 |- ( ( z e. ~P ( _V X. _V ) /\ E. x ( z = `' x /\ ( X C_ x /\ ps ) ) ) <-> E. x ( z = `' x /\ ( X C_ x /\ ps ) ) )
23	22	abbii 2567	. . . . 5 |- { z | ( z e. ~P ( _V X. _V ) /\ E. x ( z = `' x /\ ( X C_ x /\ ps ) ) ) } = { z | E. x ( z = `' x /\ ( X C_ x /\ ps ) ) }
24	9, 23	eqtri 2473	. . . 4 |- { z e. ~P ( _V X. _V ) | E. x ( z = `' x /\ ( X C_ x /\ ps ) ) } = { z | E. x ( z = `' x /\ ( X C_ x /\ ps ) ) }
25	24	inteqi 4238	. . 3 |- |^| { z e. ~P ( _V X. _V ) | E. x ( z = `' x /\ ( X C_ x /\ ps ) ) } = |^| { z | E. x ( z = `' x /\ ( X C_ x /\ ps ) ) }
26	25	a1i 11	. 2 |- ( ph -> |^| { z e. ~P ( _V X. _V ) | E. x ( z = `' x /\ ( X C_ x /\ ps ) ) } = |^| { z | E. x ( z = `' x /\ ( X C_ x /\ ps ) ) } )
27		vex 3048	. . . . . . 7 |- y e. _V
28	27	cnvex 6740	. . . . . 6 |- `' y e. _V
29	28	cnvex 6740	. . . . 5 |- `' `' y e. _V
30	29	a1i 11	. . . 4 |- ( ph -> `' `' y e. _V )
31	1, 2	ssexd 4550	. . . . . . . . . . 11 |- ( ph -> X e. _V )
32		difexg 4551	. . . . . . . . . . 11 |- ( X e. _V -> ( X \ `' `' X ) e. _V )
33	31, 32	syl 17	. . . . . . . . . 10 |- ( ph -> ( X \ `' `' X ) e. _V )
34		unexg 6592	. . . . . . . . . 10 |- ( ( `' y e. _V /\ ( X \ `' `' X ) e. _V ) -> ( `' y u. ( X \ `' `' X ) ) e. _V )
35	28, 33, 34	sylancr 669	. . . . . . . . 9 |- ( ph -> ( `' y u. ( X \ `' `' X ) ) e. _V )
36		inundif 3845	. . . . . . . . . . . . . 14 |- ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) = X
37		cnvun 5241	. . . . . . . . . . . . . . . . . . . . 21 |- `' ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) = ( `' ( X i^i `' `' X ) u. `' ( X \ `' `' X ) )
38	37	sseq1i 3456	. . . . . . . . . . . . . . . . . . . 20 |- ( `' ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) C_ y <-> ( `' ( X i^i `' `' X ) u. `' ( X \ `' `' X ) ) C_ y )
39	38	biimpi 198	. . . . . . . . . . . . . . . . . . 19 |- ( `' ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) C_ y -> ( `' ( X i^i `' `' X ) u. `' ( X \ `' `' X ) ) C_ y )
40	39	unssad 3611	. . . . . . . . . . . . . . . . . 18 |- ( `' ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) C_ y -> `' ( X i^i `' `' X ) C_ y )
41		relcnv 5207	. . . . . . . . . . . . . . . . . . . . 21 |- Rel `' `' X
42		relin2 4952	. . . . . . . . . . . . . . . . . . . . 21 |- ( Rel `' `' X -> Rel ( X i^i `' `' X ) )
43	41, 42	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 |- Rel ( X i^i `' `' X )
44		dfrel2 5286	. . . . . . . . . . . . . . . . . . . 20 |- ( Rel ( X i^i `' `' X ) <-> `' `' ( X i^i `' `' X ) = ( X i^i `' `' X ) )
45	43, 44	mpbi 212	. . . . . . . . . . . . . . . . . . 19 |- `' `' ( X i^i `' `' X ) = ( X i^i `' `' X )
46		cnvss 5007	. . . . . . . . . . . . . . . . . . 19 |- ( `' ( X i^i `' `' X ) C_ y -> `' `' ( X i^i `' `' X ) C_ `' y )
47	45, 46	syl5eqssr 3477	. . . . . . . . . . . . . . . . . 18 |- ( `' ( X i^i `' `' X ) C_ y -> ( X i^i `' `' X ) C_ `' y )
48	40, 47	syl 17	. . . . . . . . . . . . . . . . 17 |- ( `' ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) C_ y -> ( X i^i `' `' X ) C_ `' y )
49		ssid 3451	. . . . . . . . . . . . . . . . 17 |- ( X \ `' `' X ) C_ ( X \ `' `' X )
50		unss12 3606	. . . . . . . . . . . . . . . . 17 |- ( ( ( X i^i `' `' X ) C_ `' y /\ ( X \ `' `' X ) C_ ( X \ `' `' X ) ) -> ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) C_ ( `' y u. ( X \ `' `' X ) ) )
51	48, 49, 50	sylancl 668	. . . . . . . . . . . . . . . 16 |- ( `' ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) C_ y -> ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) C_ ( `' y u. ( X \ `' `' X ) ) )
52	51	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) = X -> ( `' ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) C_ y -> ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) C_ ( `' y u. ( X \ `' `' X ) ) ) )
53		cnveq 5008	. . . . . . . . . . . . . . . 16 |- ( ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) = X -> `' ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) = `' X )
54	53	sseq1d 3459	. . . . . . . . . . . . . . 15 |- ( ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) = X -> ( `' ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) C_ y <-> `' X C_ y ) )
55		sseq1 3453	. . . . . . . . . . . . . . 15 |- ( ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) = X -> ( ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) C_ ( `' y u. ( X \ `' `' X ) ) <-> X C_ ( `' y u. ( X \ `' `' X ) ) ) )
56	52, 54, 55	3imtr3d 271	. . . . . . . . . . . . . 14 |- ( ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) = X -> ( `' X C_ y -> X C_ ( `' y u. ( X \ `' `' X ) ) ) )
57	36, 56	ax-mp 5	. . . . . . . . . . . . 13 |- ( `' X C_ y -> X C_ ( `' y u. ( X \ `' `' X ) ) )
58		sseq2 3454	. . . . . . . . . . . . 13 |- ( x = ( `' y u. ( X \ `' `' X ) ) -> ( X C_ x <-> X C_ ( `' y u. ( X \ `' `' X ) ) ) )
59	57, 58	syl5ibr 225	. . . . . . . . . . . 12 |- ( x = ( `' y u. ( X \ `' `' X ) ) -> ( `' X C_ y -> X C_ x ) )
60	59	adantl 468	. . . . . . . . . . 11 |- ( ( ph /\ x = ( `' y u. ( X \ `' `' X ) ) ) -> ( `' X C_ y -> X C_ x ) )
61		clcnvlem.sub1	. . . . . . . . . . 11 |- ( ( ph /\ x = ( `' y u. ( X \ `' `' X ) ) ) -> ( ch -> ps ) )
62	60, 61	anim12d 566	. . . . . . . . . 10 |- ( ( ph /\ x = ( `' y u. ( X \ `' `' X ) ) ) -> ( ( `' X C_ y /\ ch ) -> ( X C_ x /\ ps ) ) )
63		cnveq 5008	. . . . . . . . . . . 12 |- ( x = ( `' y u. ( X \ `' `' X ) ) -> `' x = `' ( `' y u. ( X \ `' `' X ) ) )
64		cnvun 5241	. . . . . . . . . . . . 13 |- `' ( `' y u. ( X \ `' `' X ) ) = ( `' `' y u. `' ( X \ `' `' X ) )
65		cnvnonrel 36194	. . . . . . . . . . . . . . 15 |- `' ( X \ `' `' X ) = (/)
66		0ss 3763	. . . . . . . . . . . . . . 15 |- (/) C_ `' `' y
67	65, 66	eqsstri 3462	. . . . . . . . . . . . . 14 |- `' ( X \ `' `' X ) C_ `' `' y
68		ssequn2 3607	. . . . . . . . . . . . . 14 |- ( `' ( X \ `' `' X ) C_ `' `' y <-> ( `' `' y u. `' ( X \ `' `' X ) ) = `' `' y )
69	67, 68	mpbi 212	. . . . . . . . . . . . 13 |- ( `' `' y u. `' ( X \ `' `' X ) ) = `' `' y
70	64, 69	eqtr2i 2474	. . . . . . . . . . . 12 |- `' `' y = `' ( `' y u. ( X \ `' `' X ) )
71	63, 70	syl6reqr 2504	. . . . . . . . . . 11 |- ( x = ( `' y u. ( X \ `' `' X ) ) -> `' `' y = `' x )
72	71	adantl 468	. . . . . . . . . 10 |- ( ( ph /\ x = ( `' y u. ( X \ `' `' X ) ) ) -> `' `' y = `' x )
73	62, 72	jctild 546	. . . . . . . . 9 |- ( ( ph /\ x = ( `' y u. ( X \ `' `' X ) ) ) -> ( ( `' X C_ y /\ ch ) -> ( `' `' y = `' x /\ ( X C_ x /\ ps ) ) ) )
74	35, 73	spcimedv 3133	. . . . . . . 8 |- ( ph -> ( ( `' X C_ y /\ ch ) -> E. x ( `' `' y = `' x /\ ( X C_ x /\ ps ) ) ) )
75	74	imp 431	. . . . . . 7 |- ( ( ph /\ ( `' X C_ y /\ ch ) ) -> E. x ( `' `' y = `' x /\ ( X C_ x /\ ps ) ) )
76	75	adantlr 721	. . . . . 6 |- ( ( ( ph /\ z = `' `' y ) /\ ( `' X C_ y /\ ch ) ) -> E. x ( `' `' y = `' x /\ ( X C_ x /\ ps ) ) )
77		eqeq1 2455	. . . . . . . . 9 |- ( z = `' `' y -> ( z = `' x <-> `' `' y = `' x ) )
78	77	anbi1d 711	. . . . . . . 8 |- ( z = `' `' y -> ( ( z = `' x /\ ( X C_ x /\ ps ) ) <-> ( `' `' y = `' x /\ ( X C_ x /\ ps ) ) ) )
79	78	exbidv 1768	. . . . . . 7 |- ( z = `' `' y -> ( E. x ( z = `' x /\ ( X C_ x /\ ps ) ) <-> E. x ( `' `' y = `' x /\ ( X C_ x /\ ps ) ) ) )
80	79	ad2antlr 733	. . . . . 6 |- ( ( ( ph /\ z = `' `' y ) /\ ( `' X C_ y /\ ch ) ) -> ( E. x ( z = `' x /\ ( X C_ x /\ ps ) ) <-> E. x ( `' `' y = `' x /\ ( X C_ x /\ ps ) ) ) )
81	76, 80	mpbird 236	. . . . 5 |- ( ( ( ph /\ z = `' `' y ) /\ ( `' X C_ y /\ ch ) ) -> E. x ( z = `' x /\ ( X C_ x /\ ps ) ) )
82	81	ex 436	. . . 4 |- ( ( ph /\ z = `' `' y ) -> ( ( `' X C_ y /\ ch ) -> E. x ( z = `' x /\ ( X C_ x /\ ps ) ) ) )
83		cnvcnvss 5290	. . . . 5 |- `' `' y C_ y
84	83	a1i 11	. . . 4 |- ( ph -> `' `' y C_ y )
85	30, 82, 84	intabssd 36216	. . 3 |- ( ph -> |^| { z | E. x ( z = `' x /\ ( X C_ x /\ ps ) ) } C_ |^| { y | ( `' X C_ y /\ ch ) } )
86		vex 3048	. . . . 5 |- z e. _V
87	86	a1i 11	. . . 4 |- ( ph -> z e. _V )
88		eqtr 2470	. . . . . . . 8 |- ( ( y = z /\ z = `' x ) -> y = `' x )
89		cnvss 5007	. . . . . . . . . . . 12 |- ( X C_ x -> `' X C_ `' x )
90		sseq2 3454	. . . . . . . . . . . 12 |- ( y = `' x -> ( `' X C_ y <-> `' X C_ `' x ) )
91	89, 90	syl5ibr 225	. . . . . . . . . . 11 |- ( y = `' x -> ( X C_ x -> `' X C_ y ) )
92	91	adantl 468	. . . . . . . . . 10 |- ( ( ph /\ y = `' x ) -> ( X C_ x -> `' X C_ y ) )
93		clcnvlem.sub2	. . . . . . . . . 10 |- ( ( ph /\ y = `' x ) -> ( ps -> ch ) )
94	92, 93	anim12d 566	. . . . . . . . 9 |- ( ( ph /\ y = `' x ) -> ( ( X C_ x /\ ps ) -> ( `' X C_ y /\ ch ) ) )
95	94	ex 436	. . . . . . . 8 |- ( ph -> ( y = `' x -> ( ( X C_ x /\ ps ) -> ( `' X C_ y /\ ch ) ) ) )
96	88, 95	syl5 33	. . . . . . 7 |- ( ph -> ( ( y = z /\ z = `' x ) -> ( ( X C_ x /\ ps ) -> ( `' X C_ y /\ ch ) ) ) )
97	96	impl 626	. . . . . 6 |- ( ( ( ph /\ y = z ) /\ z = `' x ) -> ( ( X C_ x /\ ps ) -> ( `' X C_ y /\ ch ) ) )
98	97	expimpd 608	. . . . 5 |- ( ( ph /\ y = z ) -> ( ( z = `' x /\ ( X C_ x /\ ps ) ) -> ( `' X C_ y /\ ch ) ) )
99	98	exlimdv 1779	. . . 4 |- ( ( ph /\ y = z ) -> ( E. x ( z = `' x /\ ( X C_ x /\ ps ) ) -> ( `' X C_ y /\ ch ) ) )
100		ssid 3451	. . . . 5 |- z C_ z
101	100	a1i 11	. . . 4 |- ( ph -> z C_ z )
102	87, 99, 101	intabssd 36216	. . 3 |- ( ph -> |^| { y | ( `' X C_ y /\ ch ) } C_ |^| { z | E. x ( z = `' x /\ ( X C_ x /\ ps ) ) } )
103	85, 102	eqssd 3449	. 2 |- ( ph -> |^| { z | E. x ( z = `' x /\ ( X C_ x /\ ps ) ) } = |^| { y | ( `' X C_ y /\ ch ) } )
104	8, 26, 103	3eqtrd 2489	1 |- ( ph -> `' |^| { x | ( X C_ x /\ ps ) } = |^| { y | ( `' X C_ y /\ ch ) } )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1444   E.wex 1663   e. wcel 1887   {cab 2437   {crab 2741   _Vcvv 3045   \ cdif 3401   u. cun 3402   i^i cin 3403   C_ wss 3404   (/)c0 3731   ~Pcpw 3951   |^|cint 4234   X. cxp 4832   `'ccnv 4833   Rel wrel 4839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-int 4235  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-iota 5546  df-fun 5584  df-fv 5590  df-1st 6793  df-2nd 6794

This theorem is referenced by:  cnvtrucl0  36231  cnvrcl0  36232  cnvtrcl0  36233