Theorem opeliunxp2 4674

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 3925	. . 3 |- ( C U_ x e. A ( { x } X. B ) D <-> <. C , D >. e. U_ x e. A ( { x } X. B ) )
2		relxp 4643	. . . . . 6 |- Rel ( { x } X. B )
3	2	rgenw 2485	. . . . 5 |- A. x e. A Rel ( { x } X. B )
4		reliun 4655	. . . . 5 |- ( Rel U_ x e. A ( { x } X. B ) <-> A. x e. A Rel ( { x } X. B ) )
5	3, 4	mpbir 145	. . . 4 |- Rel U_ x e. A ( { x } X. B )
6	5	brrelex1i 4577	. . 3 |- ( C U_ x e. A ( { x } X. B ) D -> C e. _V )
7	1, 6	sylbir 134	. 2 |- ( <. C , D >. e. U_ x e. A ( { x } X. B ) -> C e. _V )
8		elex 2692	. . 3 |- ( C e. A -> C e. _V )
9	8	adantr 274	. 2 |- ( ( C e. A /\ D e. E ) -> C e. _V )
10		nfcv 2279	. . 3 |- F/_ x C
11		nfiu1 3838	. . . . 5 |- F/_ x U_ x e. A ( { x } X. B )
12	11	nfel2 2292	. . . 4 |- F/ x <. C , D >. e. U_ x e. A ( { x } X. B )
13		nfv 1508	. . . 4 |- F/ x ( C e. A /\ D e. E )
14	12, 13	nfbi 1568	. . 3 |- F/ x ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )
15		opeq1 3700	. . . . 5 |- ( x = C -> <. x , D >. = <. C , D >. )
16	15	eleq1d 2206	. . . 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 2200	. . . . 5 |- ( x = C -> ( x e. A <-> C e. A ) )
18		opeliunxp2.1	. . . . . 6 |- ( x = C -> B = E )
19	18	eleq2d 2207	. . . . 5 |- ( x = C -> ( D e. B <-> D e. E ) )
20	17, 19	anbi12d 464	. . . 4 |- ( x = C -> ( ( x e. A /\ D e. B ) <-> ( C e. A /\ D e. E ) ) )
21	16, 20	bibi12d 234	. . 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 4589	. . 3 |- ( <. x , D >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ D e. B ) )
23	10, 14, 21, 22	vtoclgf 2739	. 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 693	1 |- ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   A.wral 2414   _Vcvv 2681   {csn 3522   <.cop 3525   U_ciun 3808   class class class wbr 3924   X. cxp 4532   Rel wrel 4539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-iun 3810  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541

This theorem is referenced by:  mpoxopn0yelv  6129  eldvap  12809