Theorem opeliunxp2 4991

Description: Membership in a union of Cartesian 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 4416	. . 3 |- ( C U_ x e. A ( { x } X. B ) D <-> <. C , D >. e. U_ x e. A ( { x } X. B ) )
2		relxp 4960	. . . . . 6 |- Rel ( { x } X. B )
3	2	rgenw 2760	. . . . 5 |- A. x e. A Rel ( { x } X. B )
4		reliun 4972	. . . . 5 |- ( Rel U_ x e. A ( { x } X. B ) <-> A. x e. A Rel ( { x } X. B ) )
5	3, 4	mpbir 214	. . . 4 |- Rel U_ x e. A ( { x } X. B )
6	5	brrelexi 4893	. . 3 |- ( C U_ x e. A ( { x } X. B ) D -> C e. _V )
7	1, 6	sylbir 218	. 2 |- ( <. C , D >. e. U_ x e. A ( { x } X. B ) -> C e. _V )
8		elex 3065	. . 3 |- ( C e. A -> C e. _V )
9	8	adantr 471	. 2 |- ( ( C e. A /\ D e. E ) -> C e. _V )
10		nfiu1 4321	. . . . 5 |- F/_ x U_ x e. A ( { x } X. B )
11	10	nfel2 2618	. . . 4 |- F/ x <. C , D >. e. U_ x e. A ( { x } X. B )
12		nfv 1771	. . . 4 |- F/ x ( C e. A /\ D e. E )
13	11, 12	nfbi 2027	. . 3 |- F/ x ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )
14		opeq1 4179	. . . . 5 |- ( x = C -> <. x , D >. = <. C , D >. )
15	14	eleq1d 2523	. . . 4 |- ( x = C -> ( <. x , D >. e. U_ x e. A ( { x } X. B ) <-> <. C , D >. e. U_ x e. A ( { x } X. B ) ) )
16		eleq1 2527	. . . . 5 |- ( x = C -> ( x e. A <-> C e. A ) )
17		opeliunxp2.1	. . . . . 6 |- ( x = C -> B = E )
18	17	eleq2d 2524	. . . . 5 |- ( x = C -> ( D e. B <-> D e. E ) )
19	16, 18	anbi12d 722	. . . 4 |- ( x = C -> ( ( x e. A /\ D e. B ) <-> ( C e. A /\ D e. E ) ) )
20	15, 19	bibi12d 327	. . 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 ) ) ) )
21		opeliunxp 4904	. . 3 |- ( <. x , D >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ D e. B ) )
22	13, 20, 21	vtoclg1f 3117	. 2 |- ( C e. _V -> ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) ) )
23	7, 9, 22	pm5.21nii 359	1 |- ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1454   e. wcel 1897   A.wral 2748   _Vcvv 3056   {csn 3979   <.cop 3985   U_ciun 4291   class class class wbr 4415   X. cxp 4850   Rel wrel 4857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-iun 4293  df-br 4416  df-opab 4475  df-xp 4858  df-rel 4859

This theorem is referenced by:  mpt2xopn0yelv  6985  mpt2xopxnop0  6987  eldmcoa  16008  dmdprd  17678  ply1frcl  18955  cnextfres  21132  eldv  22901  perfdvf  22906  eltayl  23363  dfcnv2  28327  cvmliftlem1  30056  filnetlem3  31084