Theorem opeliunxp2 4494

Description: Membership in a union of cross products. (Contributed by Mario Carneiro, 14-Feb-2015.)

Hypothesis

Ref	Expression
opeliunxp2.1	|- ( x = C -> B = E )

Assertion

Ref	Expression
opeliunxp2	|- ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )

Distinct variable groups:   x, C   x, D   x, E   x, A

Allowed substitution hint:   B( x)

Proof of Theorem opeliunxp2

Step	Hyp	Ref	Expression
1		df-br 3786	. . 3 |- ( C U_ x e. A ( { x } X. B ) D <-> <. C , D >. e. U_ x e. A ( { x } X. B ) )
2		relxp 4465	. . . . . 6 |- Rel ( { x } X. B )
3	2	rgenw 2418	. . . . 5 |- A. x e. A Rel ( { x } X. B )
4		reliun 4476	. . . . 5 |- ( Rel U_ x e. A ( { x } X. B ) <-> A. x e. A Rel ( { x } X. B ) )
5	3, 4	mpbir 144	. . . 4 |- Rel U_ x e. A ( { x } X. B )
6	5	brrelexi 4402	. . 3 |- ( C U_ x e. A ( { x } X. B ) D -> C e. _V )
7	1, 6	sylbir 133	. 2 |- ( <. C , D >. e. U_ x e. A ( { x } X. B ) -> C e. _V )
8		elex 2610	. . 3 |- ( C e. A -> C e. _V )
9	8	adantr 270	. 2 |- ( ( C e. A /\ D e. E ) -> C e. _V )
10		nfcv 2219	. . 3 |- F/_ x C
11		nfiu1 3708	. . . . 5 |- F/_ x U_ x e. A ( { x } X. B )
12	11	nfel2 2231	. . . 4 |- F/ x <. C , D >. e. U_ x e. A ( { x } X. B )
13		nfv 1461	. . . 4 |- F/ x ( C e. A /\ D e. E )
14	12, 13	nfbi 1521	. . 3 |- F/ x ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )
15		opeq1 3570	. . . . 5 |- ( x = C -> <. x , D >. = <. C , D >. )
16	15	eleq1d 2147	. . . 4 |- ( x = C -> ( <. x , D >. e. U_ x e. A ( { x } X. B ) <-> <. C , D >. e. U_ x e. A ( { x } X. B ) ) )
17		eleq1 2141	. . . . 5 |- ( x = C -> ( x e. A <-> C e. A ) )
18		opeliunxp2.1	. . . . . 6 |- ( x = C -> B = E )
19	18	eleq2d 2148	. . . . 5 |- ( x = C -> ( D e. B <-> D e. E ) )
20	17, 19	anbi12d 456	. . . 4 |- ( x = C -> ( ( x e. A /\ D e. B ) <-> ( C e. A /\ D e. E ) ) )
21	16, 20	bibi12d 233	. . 3 |- ( x = C -> ( ( <. x , D >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ D e. B ) ) <-> ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) ) ) )
22		opeliunxp 4413	. . 3 |- ( <. x , D >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ D e. B ) )
23	10, 14, 21, 22	vtoclgf 2657	. 2 |- ( C e. _V -> ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) ) )
24	7, 9, 23	pm5.21nii 652	1 |- ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348   _Vcvv 2601   {csn 3398   <.cop 3401   U_ciun 3678   class class class wbr 3785   X. cxp 4361   Rel wrel 4368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-iun 3680  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370

This theorem is referenced by:  mpt2xopn0yelv  5877