Theorem cnviinm 5088

Description: The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.)

Assertion

Ref	Expression
cnviinm	|- ( E. y y e. A -> `' |^|_ x e. A B = |^|_ x e. A `' B )

Distinct variable groups:   x, A   y, A

Allowed substitution hints:   B( x, y)

Proof of Theorem cnviinm

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

Step	Hyp	Ref	Expression
1		eleq1w 2201	. . 3 |- ( y = a -> ( y e. A <-> a e. A ) )
2	1	cbvexv 1891	. 2 |- ( E. y y e. A <-> E. a a e. A )
3		eleq1w 2201	. . . 4 |- ( x = a -> ( x e. A <-> a e. A ) )
4	3	cbvexv 1891	. . 3 |- ( E. x x e. A <-> E. a a e. A )
5		relcnv 4925	. . . 4 |- Rel `' |^|_ x e. A B
6		r19.2m 3454	. . . . . . . 8 |- ( ( E. x x e. A /\ A. x e. A `' B C_ ( _V X. _V ) ) -> E. x e. A `' B C_ ( _V X. _V ) )
7	6	expcom 115	. . . . . . 7 |- ( A. x e. A `' B C_ ( _V X. _V ) -> ( E. x x e. A -> E. x e. A `' B C_ ( _V X. _V ) ) )
8		relcnv 4925	. . . . . . . . 9 |- Rel `' B
9		df-rel 4554	. . . . . . . . 9 |- ( Rel `' B <-> `' B C_ ( _V X. _V ) )
10	8, 9	mpbi 144	. . . . . . . 8 |- `' B C_ ( _V X. _V )
11	10	a1i 9	. . . . . . 7 |- ( x e. A -> `' B C_ ( _V X. _V ) )
12	7, 11	mprg 2492	. . . . . 6 |- ( E. x x e. A -> E. x e. A `' B C_ ( _V X. _V ) )
13		iinss 3872	. . . . . 6 |- ( E. x e. A `' B C_ ( _V X. _V ) -> |^|_ x e. A `' B C_ ( _V X. _V ) )
14	12, 13	syl 14	. . . . 5 |- ( E. x x e. A -> |^|_ x e. A `' B C_ ( _V X. _V ) )
15		df-rel 4554	. . . . 5 |- ( Rel |^|_ x e. A `' B <-> |^|_ x e. A `' B C_ ( _V X. _V ) )
16	14, 15	sylibr 133	. . . 4 |- ( E. x x e. A -> Rel |^|_ x e. A `' B )
17		vex 2692	. . . . . . . 8 |- b e. _V
18		vex 2692	. . . . . . . 8 |- a e. _V
19	17, 18	opex 4159	. . . . . . 7 |- <. b , a >. e. _V
20		eliin 3826	. . . . . . 7 |- ( <. b , a >. e. _V -> ( <. b , a >. e. |^|_ x e. A B <-> A. x e. A <. b , a >. e. B ) )
21	19, 20	ax-mp 5	. . . . . 6 |- ( <. b , a >. e. |^|_ x e. A B <-> A. x e. A <. b , a >. e. B )
22	18, 17	opelcnv 4729	. . . . . 6 |- ( <. a , b >. e. `' |^|_ x e. A B <-> <. b , a >. e. |^|_ x e. A B )
23	18, 17	opex 4159	. . . . . . . 8 |- <. a , b >. e. _V
24		eliin 3826	. . . . . . . 8 |- ( <. a , b >. e. _V -> ( <. a , b >. e. |^|_ x e. A `' B <-> A. x e. A <. a , b >. e. `' B ) )
25	23, 24	ax-mp 5	. . . . . . 7 |- ( <. a , b >. e. |^|_ x e. A `' B <-> A. x e. A <. a , b >. e. `' B )
26	18, 17	opelcnv 4729	. . . . . . . 8 |- ( <. a , b >. e. `' B <-> <. b , a >. e. B )
27	26	ralbii 2444	. . . . . . 7 |- ( A. x e. A <. a , b >. e. `' B <-> A. x e. A <. b , a >. e. B )
28	25, 27	bitri 183	. . . . . 6 |- ( <. a , b >. e. |^|_ x e. A `' B <-> A. x e. A <. b , a >. e. B )
29	21, 22, 28	3bitr4i 211	. . . . 5 |- ( <. a , b >. e. `' |^|_ x e. A B <-> <. a , b >. e. |^|_ x e. A `' B )
30	29	eqrelriv 4640	. . . 4 |- ( ( Rel `' |^|_ x e. A B /\ Rel |^|_ x e. A `' B ) -> `' |^|_ x e. A B = |^|_ x e. A `' B )
31	5, 16, 30	sylancr 411	. . 3 |- ( E. x x e. A -> `' |^|_ x e. A B = |^|_ x e. A `' B )
32	4, 31	sylbir 134	. 2 |- ( E. a a e. A -> `' |^|_ x e. A B = |^|_ x e. A `' B )
33	2, 32	sylbi 120	1 |- ( E. y y e. A -> `' |^|_ x e. A B = |^|_ x e. A `' B )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   E.wex 1469   e. wcel 1481   A.wral 2417   E.wrex 2418   _Vcvv 2689   C_ wss 3076   <.cop 3535   |^|_ciin 3822   X. cxp 4545   `'ccnv 4546   Rel wrel 4552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-iin 3824  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555

This theorem is referenced by: (None)