Theorem cnviinm 5270

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 2290	. . 3 |- ( y = a -> ( y e. A <-> a e. A ) )
2	1	cbvexv 1965	. 2 |- ( E. y y e. A <-> E. a a e. A )
3		eleq1w 2290	. . . 4 |- ( x = a -> ( x e. A <-> a e. A ) )
4	3	cbvexv 1965	. . 3 |- ( E. x x e. A <-> E. a a e. A )
5		relcnv 5106	. . . 4 |- Rel `' |^|_ x e. A B
6		r19.2m 3578	. . . . . . . 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 5106	. . . . . . . . 9 |- Rel `' B
9		df-rel 4726	. . . . . . . . 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 2587	. . . . . 6 |- ( E. x x e. A -> E. x e. A `' B C_ ( _V X. _V ) )
13		iinss 4017	. . . . . 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 4726	. . . . 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 2802	. . . . . . . 8 |- b e. _V
18		vex 2802	. . . . . . . 8 |- a e. _V
19	17, 18	opex 4315	. . . . . . 7 |- <. b , a >. e. _V
20		eliin 3970	. . . . . . 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 4904	. . . . . 6 |- ( <. a , b >. e. `' |^|_ x e. A B <-> <. b , a >. e. |^|_ x e. A B )
23	18, 17	opex 4315	. . . . . . . 8 |- <. a , b >. e. _V
24		eliin 3970	. . . . . . . 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 4904	. . . . . . . 8 |- ( <. a , b >. e. `' B <-> <. b , a >. e. B )
27	26	ralbii 2536	. . . . . . 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 4812	. . . 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 1395   E.wex 1538   e. wcel 2200   A.wral 2508   E.wrex 2509   _Vcvv 2799   C_ wss 3197   <.cop 3669   |^|_ciin 3966   X. cxp 4717   `'ccnv 4718   Rel wrel 4724

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-iin 3968  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727

This theorem is referenced by: (None)