Theorem cnviinm 5207

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 2254	. . 3 |- ( y = a -> ( y e. A <-> a e. A ) )
2	1	cbvexv 1930	. 2 |- ( E. y y e. A <-> E. a a e. A )
3		eleq1w 2254	. . . 4 |- ( x = a -> ( x e. A <-> a e. A ) )
4	3	cbvexv 1930	. . 3 |- ( E. x x e. A <-> E. a a e. A )
5		relcnv 5043	. . . 4 |- Rel `' |^|_ x e. A B
6		r19.2m 3533	. . . . . . . 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 116	. . . . . . 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 5043	. . . . . . . . 9 |- Rel `' B
9		df-rel 4666	. . . . . . . . 9 |- ( Rel `' B <-> `' B C_ ( _V X. _V ) )
10	8, 9	mpbi 145	. . . . . . . 8 |- `' B C_ ( _V X. _V )
11	10	a1i 9	. . . . . . 7 |- ( x e. A -> `' B C_ ( _V X. _V ) )
12	7, 11	mprg 2551	. . . . . 6 |- ( E. x x e. A -> E. x e. A `' B C_ ( _V X. _V ) )
13		iinss 3964	. . . . . 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 4666	. . . . 5 |- ( Rel |^|_ x e. A `' B <-> |^|_ x e. A `' B C_ ( _V X. _V ) )
16	14, 15	sylibr 134	. . . 4 |- ( E. x x e. A -> Rel |^|_ x e. A `' B )
17		vex 2763	. . . . . . . 8 |- b e. _V
18		vex 2763	. . . . . . . 8 |- a e. _V
19	17, 18	opex 4258	. . . . . . 7 |- <. b , a >. e. _V
20		eliin 3917	. . . . . . 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 4844	. . . . . 6 |- ( <. a , b >. e. `' |^|_ x e. A B <-> <. b , a >. e. |^|_ x e. A B )
23	18, 17	opex 4258	. . . . . . . 8 |- <. a , b >. e. _V
24		eliin 3917	. . . . . . . 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 4844	. . . . . . . 8 |- ( <. a , b >. e. `' B <-> <. b , a >. e. B )
27	26	ralbii 2500	. . . . . . 7 |- ( A. x e. A <. a , b >. e. `' B <-> A. x e. A <. b , a >. e. B )
28	25, 27	bitri 184	. . . . . 6 |- ( <. a , b >. e. |^|_ x e. A `' B <-> A. x e. A <. b , a >. e. B )
29	21, 22, 28	3bitr4i 212	. . . . 5 |- ( <. a , b >. e. `' |^|_ x e. A B <-> <. a , b >. e. |^|_ x e. A `' B )
30	29	eqrelriv 4752	. . . 4 |- ( ( Rel `' |^|_ x e. A B /\ Rel |^|_ x e. A `' B ) -> `' |^|_ x e. A B = |^|_ x e. A `' B )
31	5, 16, 30	sylancr 414	. . 3 |- ( E. x x e. A -> `' |^|_ x e. A B = |^|_ x e. A `' B )
32	4, 31	sylbir 135	. 2 |- ( E. a a e. A -> `' |^|_ x e. A B = |^|_ x e. A `' B )
33	2, 32	sylbi 121	1 |- ( E. y y e. A -> `' |^|_ x e. A B = |^|_ x e. A `' B )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   E.wex 1503   e. wcel 2164   A.wral 2472   E.wrex 2473   _Vcvv 2760   C_ wss 3153   <.cop 3621   |^|_ciin 3913   X. cxp 4657   `'ccnv 4658   Rel wrel 4664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-iin 3915  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667

This theorem is referenced by: (None)