Theorem cnviin 5672

Description: The converse of an intersection is the intersection of the converse. (Contributed by FL, 15-Oct-2012.)

Assertion

Ref	Expression
cnviin	|- ( A =/= (/) -> `' |^|_ x e. A B = |^|_ x e. A `' B )

Distinct variable group:   x, A

Allowed substitution hint:   B( x)

Proof of Theorem cnviin

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcnv 5503	. 2 |- Rel `' |^|_ x e. A B
2		relcnv 5503	. . . . . . 7 |- Rel `' B
3		df-rel 5121	. . . . . . 7 |- ( Rel `' B <-> `' B C_ ( _V X. _V ) )
4	2, 3	mpbi 220	. . . . . 6 |- `' B C_ ( _V X. _V )
5	4	rgenw 2924	. . . . 5 |- A. x e. A `' B C_ ( _V X. _V )
6		r19.2z 4060	. . . . 5 |- ( ( A =/= (/) /\ A. x e. A `' B C_ ( _V X. _V ) ) -> E. x e. A `' B C_ ( _V X. _V ) )
7	5, 6	mpan2 707	. . . 4 |- ( A =/= (/) -> E. x e. A `' B C_ ( _V X. _V ) )
8		iinss 4571	. . . 4 |- ( E. x e. A `' B C_ ( _V X. _V ) -> |^|_ x e. A `' B C_ ( _V X. _V ) )
9	7, 8	syl 17	. . 3 |- ( A =/= (/) -> |^|_ x e. A `' B C_ ( _V X. _V ) )
10		df-rel 5121	. . 3 |- ( Rel |^|_ x e. A `' B <-> |^|_ x e. A `' B C_ ( _V X. _V ) )
11	9, 10	sylibr 224	. 2 |- ( A =/= (/) -> Rel |^|_ x e. A `' B )
12		opex 4932	. . . . 5 |- <. b , a >. e. _V
13		eliin 4525	. . . . 5 |- ( <. b , a >. e. _V -> ( <. b , a >. e. |^|_ x e. A B <-> A. x e. A <. b , a >. e. B ) )
14	12, 13	ax-mp 5	. . . 4 |- ( <. b , a >. e. |^|_ x e. A B <-> A. x e. A <. b , a >. e. B )
15		vex 3203	. . . . 5 |- a e. _V
16		vex 3203	. . . . 5 |- b e. _V
17	15, 16	opelcnv 5304	. . . 4 |- ( <. a , b >. e. `' |^|_ x e. A B <-> <. b , a >. e. |^|_ x e. A B )
18		opex 4932	. . . . . 6 |- <. a , b >. e. _V
19		eliin 4525	. . . . . 6 |- ( <. a , b >. e. _V -> ( <. a , b >. e. |^|_ x e. A `' B <-> A. x e. A <. a , b >. e. `' B ) )
20	18, 19	ax-mp 5	. . . . 5 |- ( <. a , b >. e. |^|_ x e. A `' B <-> A. x e. A <. a , b >. e. `' B )
21	15, 16	opelcnv 5304	. . . . . 6 |- ( <. a , b >. e. `' B <-> <. b , a >. e. B )
22	21	ralbii 2980	. . . . 5 |- ( A. x e. A <. a , b >. e. `' B <-> A. x e. A <. b , a >. e. B )
23	20, 22	bitri 264	. . . 4 |- ( <. a , b >. e. |^|_ x e. A `' B <-> A. x e. A <. b , a >. e. B )
24	14, 17, 23	3bitr4i 292	. . 3 |- ( <. a , b >. e. `' |^|_ x e. A B <-> <. a , b >. e. |^|_ x e. A `' B )
25	24	eqrelriv 5213	. 2 |- ( ( Rel `' |^|_ x e. A B /\ Rel |^|_ x e. A `' B ) -> `' |^|_ x e. A B = |^|_ x e. A `' B )
26	1, 11, 25	sylancr 695	1 |- ( A =/= (/) -> `' |^|_ x e. A B = |^|_ x e. A `' B )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   <.cop 4183   |^|_ciin 4521   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-iin 4523  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122

This theorem is referenced by: (None)