Theorem cnviinm 5080

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 2200	. . 3 |- ( y = a -> ( y e. A <-> a e. A ) )
2	1	cbvexv 1890	. 2 |- ( E. y y e. A <-> E. a a e. A )
3		eleq1w 2200	. . . 4 |- ( x = a -> ( x e. A <-> a e. A ) )
4	3	cbvexv 1890	. . 3 |- ( E. x x e. A <-> E. a a e. A )
5		relcnv 4917	. . . 4 |- Rel `' |^|_ x e. A B
6		r19.2m 3449	. . . . . . . 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 4917	. . . . . . . . 9 |- Rel `' B
9		df-rel 4546	. . . . . . . . 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 2489	. . . . . 6 |- ( E. x x e. A -> E. x e. A `' B C_ ( _V X. _V ) )
13		iinss 3864	. . . . . 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 4546	. . . . 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 2689	. . . . . . . 8 |- b e. _V
18		vex 2689	. . . . . . . 8 |- a e. _V
19	17, 18	opex 4151	. . . . . . 7 |- <. b , a >. e. _V
20		eliin 3818	. . . . . . 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 4721	. . . . . 6 |- ( <. a , b >. e. `' |^|_ x e. A B <-> <. b , a >. e. |^|_ x e. A B )
23	18, 17	opex 4151	. . . . . . . 8 |- <. a , b >. e. _V
24		eliin 3818	. . . . . . . 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 4721	. . . . . . . 8 |- ( <. a , b >. e. `' B <-> <. b , a >. e. B )
27	26	ralbii 2441	. . . . . . 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 4632	. . . 4 |- ( ( Rel `' |^|_ x e. A B /\ Rel |^|_ x e. A `' B ) -> `' |^|_ x e. A B = |^|_ x e. A `' B )
31	5, 16, 30	sylancr 410	. . 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 1331   E.wex 1468   e. wcel 1480   A.wral 2416   E.wrex 2417   _Vcvv 2686   C_ wss 3071   <.cop 3530   |^|_ciin 3814   X. cxp 4537   `'ccnv 4538   Rel wrel 4544

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-iin 3816  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547

This theorem is referenced by: (None)