Theorem opeliunxp2 5260

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 4654	. . 3 |- ( C U_ x e. A ( { x } X. B ) D <-> <. C , D >. e. U_ x e. A ( { x } X. B ) )
2		relxp 5227	. . . . . 6 |- Rel ( { x } X. B )
3	2	rgenw 2924	. . . . 5 |- A. x e. A Rel ( { x } X. B )
4		reliun 5239	. . . . 5 |- ( Rel U_ x e. A ( { x } X. B ) <-> A. x e. A Rel ( { x } X. B ) )
5	3, 4	mpbir 221	. . . 4 |- Rel U_ x e. A ( { x } X. B )
6	5	brrelexi 5158	. . 3 |- ( C U_ x e. A ( { x } X. B ) D -> C e. _V )
7	1, 6	sylbir 225	. 2 |- ( <. C , D >. e. U_ x e. A ( { x } X. B ) -> C e. _V )
8		elex 3212	. . 3 |- ( C e. A -> C e. _V )
9	8	adantr 481	. 2 |- ( ( C e. A /\ D e. E ) -> C e. _V )
10		nfiu1 4550	. . . . 5 |- F/_ x U_ x e. A ( { x } X. B )
11	10	nfel2 2781	. . . 4 |- F/ x <. C , D >. e. U_ x e. A ( { x } X. B )
12		nfv 1843	. . . 4 |- F/ x ( C e. A /\ D e. E )
13	11, 12	nfbi 1833	. . 3 |- F/ x ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )
14		opeq1 4402	. . . . 5 |- ( x = C -> <. x , D >. = <. C , D >. )
15	14	eleq1d 2686	. . . 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 2689	. . . . 5 |- ( x = C -> ( x e. A <-> C e. A ) )
17		opeliunxp2.1	. . . . . 6 |- ( x = C -> B = E )
18	17	eleq2d 2687	. . . . 5 |- ( x = C -> ( D e. B <-> D e. E ) )
19	16, 18	anbi12d 747	. . . 4 |- ( x = C -> ( ( x e. A /\ D e. B ) <-> ( C e. A /\ D e. E ) ) )
20	15, 19	bibi12d 335	. . 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 5170	. . 3 |- ( <. x , D >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ D e. B ) )
22	13, 20, 21	vtoclg1f 3265	. 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 368	1 |- ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   {csn 4177   <.cop 4183   U_ciun 4520   class class class wbr 4653   X. cxp 5112   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-iun 4522  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  mpt2xopn0yelv  7339  mpt2xopxnop0  7341  eldmcoa  16715  dmdprd  18397  ply1frcl  19683  cnextfres  21873  eldv  23662  perfdvf  23667  eltayl  24114  dfcnv2  29476  cvmliftlem1  31267  filnetlem3  32375