Theorem clcnvlem 37930

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 554	. . . 4 |- ( ph -> ( X C_ A /\ th ) )
5		clcnvlem.sub3	. . . . 5 |- ( x = A -> ( ps <-> th ) )
6	5	cleq2lem 37914	. . . 4 |- ( x = A -> ( ( X C_ x /\ ps ) <-> ( X C_ A /\ th ) ) )
7	1, 4, 6	elabd 3352	. . 3 |- ( ph -> E. x ( X C_ x /\ ps ) )
8	7	cnvintabd 37909	. 2 |- ( ph -> `' |^| { x | ( X C_ x /\ ps ) } = |^| { z e. ~P ( _V X. _V ) | E. x ( z = `' x /\ ( X C_ x /\ ps ) ) } )
9		df-rab 2921	. . . . 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 1795	. . . . . . . . . . 11 |- ( E. x ( z = `' x /\ ( X C_ x /\ ps ) ) -> E. x z = `' x )
11		relcnv 5503	. . . . . . . . . . . . 13 |- Rel `' x
12		releq 5201	. . . . . . . . . . . . 13 |- ( z = `' x -> ( Rel z <-> Rel `' x ) )
13	11, 12	mpbiri 248	. . . . . . . . . . . 12 |- ( z = `' x -> Rel z )
14	13	exlimiv 1858	. . . . . . . . . . 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 5121	. . . . . . . . . 10 |- ( Rel z <-> z C_ ( _V X. _V ) )
17	15, 16	sylib 208	. . . . . . . . 9 |- ( E. x ( z = `' x /\ ( X C_ x /\ ps ) ) -> z C_ ( _V X. _V ) )
18		selpw 4165	. . . . . . . . . 10 |- ( z e. ~P ( _V X. _V ) <-> z C_ ( _V X. _V ) )
19	18	bicomi 214	. . . . . . . . 9 |- ( z C_ ( _V X. _V ) <-> z e. ~P ( _V X. _V ) )
20	17, 19	sylib 208	. . . . . . . 8 |- ( E. x ( z = `' x /\ ( X C_ x /\ ps ) ) -> z e. ~P ( _V X. _V ) )
21	20	pm4.71ri 665	. . . . . . 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 214	. . . . . 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 2739	. . . . 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 2644	. . . 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 4479	. . 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 3203	. . . . . . 7 |- y e. _V
28	27	cnvex 7113	. . . . . 6 |- `' y e. _V
29	28	cnvex 7113	. . . . 5 |- `' `' y e. _V
30	29	a1i 11	. . . 4 |- ( ph -> `' `' y e. _V )
31	1, 2	ssexd 4805	. . . . . . . . . . 11 |- ( ph -> X e. _V )
32		difexg 4808	. . . . . . . . . . 11 |- ( X e. _V -> ( X \ `' `' X ) e. _V )
33	31, 32	syl 17	. . . . . . . . . 10 |- ( ph -> ( X \ `' `' X ) e. _V )
34		unexg 6959	. . . . . . . . . 10 |- ( ( `' y e. _V /\ ( X \ `' `' X ) e. _V ) -> ( `' y u. ( X \ `' `' X ) ) e. _V )
35	28, 33, 34	sylancr 695	. . . . . . . . 9 |- ( ph -> ( `' y u. ( X \ `' `' X ) ) e. _V )
36		inundif 4046	. . . . . . . . . . . . . 14 |- ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) = X
37		cnvun 5538	. . . . . . . . . . . . . . . . . . . . 21 |- `' ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) = ( `' ( X i^i `' `' X ) u. `' ( X \ `' `' X ) )
38	37	sseq1i 3629	. . . . . . . . . . . . . . . . . . . 20 |- ( `' ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) C_ y <-> ( `' ( X i^i `' `' X ) u. `' ( X \ `' `' X ) ) C_ y )
39	38	biimpi 206	. . . . . . . . . . . . . . . . . . 19 |- ( `' ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) C_ y -> ( `' ( X i^i `' `' X ) u. `' ( X \ `' `' X ) ) C_ y )
40	39	unssad 3790	. . . . . . . . . . . . . . . . . 18 |- ( `' ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) C_ y -> `' ( X i^i `' `' X ) C_ y )
41		relcnv 5503	. . . . . . . . . . . . . . . . . . . . 21 |- Rel `' `' X
42		relin2 5237	. . . . . . . . . . . . . . . . . . . . 21 |- ( Rel `' `' X -> Rel ( X i^i `' `' X ) )
43	41, 42	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 |- Rel ( X i^i `' `' X )
44		dfrel2 5583	. . . . . . . . . . . . . . . . . . . 20 |- ( Rel ( X i^i `' `' X ) <-> `' `' ( X i^i `' `' X ) = ( X i^i `' `' X ) )
45	43, 44	mpbi 220	. . . . . . . . . . . . . . . . . . 19 |- `' `' ( X i^i `' `' X ) = ( X i^i `' `' X )
46		cnvss 5294	. . . . . . . . . . . . . . . . . . 19 |- ( `' ( X i^i `' `' X ) C_ y -> `' `' ( X i^i `' `' X ) C_ `' y )
47	45, 46	syl5eqssr 3650	. . . . . . . . . . . . . . . . . 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 3624	. . . . . . . . . . . . . . . . 17 |- ( X \ `' `' X ) C_ ( X \ `' `' X )
50		unss12 3785	. . . . . . . . . . . . . . . . 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 694	. . . . . . . . . . . . . . . 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 5296	. . . . . . . . . . . . . . . 16 |- ( ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) = X -> `' ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) = `' X )
54	53	sseq1d 3632	. . . . . . . . . . . . . . 15 |- ( ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) = X -> ( `' ( ( X i^i `' `' X ) u. ( X \ `' `' X ) ) C_ y <-> `' X C_ y ) )
55		sseq1 3626	. . . . . . . . . . . . . . 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 282	. . . . . . . . . . . . . 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 3627	. . . . . . . . . . . . 13 |- ( x = ( `' y u. ( X \ `' `' X ) ) -> ( X C_ x <-> X C_ ( `' y u. ( X \ `' `' X ) ) ) )
59	57, 58	syl5ibr 236	. . . . . . . . . . . 12 |- ( x = ( `' y u. ( X \ `' `' X ) ) -> ( `' X C_ y -> X C_ x ) )
60	59	adantl 482	. . . . . . . . . . 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 586	. . . . . . . . . 10 |- ( ( ph /\ x = ( `' y u. ( X \ `' `' X ) ) ) -> ( ( `' X C_ y /\ ch ) -> ( X C_ x /\ ps ) ) )
63		cnveq 5296	. . . . . . . . . . . 12 |- ( x = ( `' y u. ( X \ `' `' X ) ) -> `' x = `' ( `' y u. ( X \ `' `' X ) ) )
64		cnvun 5538	. . . . . . . . . . . . 13 |- `' ( `' y u. ( X \ `' `' X ) ) = ( `' `' y u. `' ( X \ `' `' X ) )
65		cnvnonrel 37894	. . . . . . . . . . . . . . 15 |- `' ( X \ `' `' X ) = (/)
66		0ss 3972	. . . . . . . . . . . . . . 15 |- (/) C_ `' `' y
67	65, 66	eqsstri 3635	. . . . . . . . . . . . . 14 |- `' ( X \ `' `' X ) C_ `' `' y
68		ssequn2 3786	. . . . . . . . . . . . . 14 |- ( `' ( X \ `' `' X ) C_ `' `' y <-> ( `' `' y u. `' ( X \ `' `' X ) ) = `' `' y )
69	67, 68	mpbi 220	. . . . . . . . . . . . 13 |- ( `' `' y u. `' ( X \ `' `' X ) ) = `' `' y
70	64, 69	eqtr2i 2645	. . . . . . . . . . . 12 |- `' `' y = `' ( `' y u. ( X \ `' `' X ) )
71	63, 70	syl6reqr 2675	. . . . . . . . . . 11 |- ( x = ( `' y u. ( X \ `' `' X ) ) -> `' `' y = `' x )
72	71	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ x = ( `' y u. ( X \ `' `' X ) ) ) -> `' `' y = `' x )
73	62, 72	jctild 566	. . . . . . . . 9 |- ( ( ph /\ x = ( `' y u. ( X \ `' `' X ) ) ) -> ( ( `' X C_ y /\ ch ) -> ( `' `' y = `' x /\ ( X C_ x /\ ps ) ) ) )
74	35, 73	spcimedv 3292	. . . . . . . 8 |- ( ph -> ( ( `' X C_ y /\ ch ) -> E. x ( `' `' y = `' x /\ ( X C_ x /\ ps ) ) ) )
75	74	imp 445	. . . . . . 7 |- ( ( ph /\ ( `' X C_ y /\ ch ) ) -> E. x ( `' `' y = `' x /\ ( X C_ x /\ ps ) ) )
76	75	adantlr 751	. . . . . 6 |- ( ( ( ph /\ z = `' `' y ) /\ ( `' X C_ y /\ ch ) ) -> E. x ( `' `' y = `' x /\ ( X C_ x /\ ps ) ) )
77		eqeq1 2626	. . . . . . . . 9 |- ( z = `' `' y -> ( z = `' x <-> `' `' y = `' x ) )
78	77	anbi1d 741	. . . . . . . 8 |- ( z = `' `' y -> ( ( z = `' x /\ ( X C_ x /\ ps ) ) <-> ( `' `' y = `' x /\ ( X C_ x /\ ps ) ) ) )
79	78	exbidv 1850	. . . . . . 7 |- ( z = `' `' y -> ( E. x ( z = `' x /\ ( X C_ x /\ ps ) ) <-> E. x ( `' `' y = `' x /\ ( X C_ x /\ ps ) ) ) )
80	79	ad2antlr 763	. . . . . 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 247	. . . . 5 |- ( ( ( ph /\ z = `' `' y ) /\ ( `' X C_ y /\ ch ) ) -> E. x ( z = `' x /\ ( X C_ x /\ ps ) ) )
82	81	ex 450	. . . 4 |- ( ( ph /\ z = `' `' y ) -> ( ( `' X C_ y /\ ch ) -> E. x ( z = `' x /\ ( X C_ x /\ ps ) ) ) )
83		cnvcnvss 5589	. . . . 5 |- `' `' y C_ y
84	83	a1i 11	. . . 4 |- ( ph -> `' `' y C_ y )
85	30, 82, 84	intabssd 37916	. . 3 |- ( ph -> |^| { z | E. x ( z = `' x /\ ( X C_ x /\ ps ) ) } C_ |^| { y | ( `' X C_ y /\ ch ) } )
86		vex 3203	. . . . 5 |- z e. _V
87	86	a1i 11	. . . 4 |- ( ph -> z e. _V )
88		eqtr 2641	. . . . . . . 8 |- ( ( y = z /\ z = `' x ) -> y = `' x )
89		cnvss 5294	. . . . . . . . . . . 12 |- ( X C_ x -> `' X C_ `' x )
90		sseq2 3627	. . . . . . . . . . . 12 |- ( y = `' x -> ( `' X C_ y <-> `' X C_ `' x ) )
91	89, 90	syl5ibr 236	. . . . . . . . . . 11 |- ( y = `' x -> ( X C_ x -> `' X C_ y ) )
92	91	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ y = `' x ) -> ( X C_ x -> `' X C_ y ) )
93		clcnvlem.sub2	. . . . . . . . . 10 |- ( ( ph /\ y = `' x ) -> ( ps -> ch ) )
94	92, 93	anim12d 586	. . . . . . . . 9 |- ( ( ph /\ y = `' x ) -> ( ( X C_ x /\ ps ) -> ( `' X C_ y /\ ch ) ) )
95	94	ex 450	. . . . . . . 8 |- ( ph -> ( y = `' x -> ( ( X C_ x /\ ps ) -> ( `' X C_ y /\ ch ) ) ) )
96	88, 95	syl5 34	. . . . . . 7 |- ( ph -> ( ( y = z /\ z = `' x ) -> ( ( X C_ x /\ ps ) -> ( `' X C_ y /\ ch ) ) ) )
97	96	impl 650	. . . . . 6 |- ( ( ( ph /\ y = z ) /\ z = `' x ) -> ( ( X C_ x /\ ps ) -> ( `' X C_ y /\ ch ) ) )
98	97	expimpd 629	. . . . 5 |- ( ( ph /\ y = z ) -> ( ( z = `' x /\ ( X C_ x /\ ps ) ) -> ( `' X C_ y /\ ch ) ) )
99	98	exlimdv 1861	. . . 4 |- ( ( ph /\ y = z ) -> ( E. x ( z = `' x /\ ( X C_ x /\ ps ) ) -> ( `' X C_ y /\ ch ) ) )
100		ssid 3624	. . . . 5 |- z C_ z
101	100	a1i 11	. . . 4 |- ( ph -> z C_ z )
102	87, 99, 101	intabssd 37916	. . 3 |- ( ph -> |^| { y | ( `' X C_ y /\ ch ) } C_ |^| { z | E. x ( z = `' x /\ ( X C_ x /\ ps ) ) } )
103	85, 102	eqssd 3620	. 2 |- ( ph -> |^| { z | E. x ( z = `' x /\ ( X C_ x /\ ps ) ) } = |^| { y | ( `' X C_ y /\ ch ) } )
104	8, 26, 103	3eqtrd 2660	1 |- ( ph -> `' |^| { x | ( X C_ x /\ ps ) } = |^| { y | ( `' X C_ y /\ ch ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   {crab 2916   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   |^|cint 4475   X. cxp 5112   `'ccnv 5113   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  cnvtrucl0  37931  cnvrcl0  37932  cnvtrcl0  37933