Theorem opeliunxp2 4639

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 3896	. . 3 |- ( C U_ x e. A ( { x } X. B ) D <-> <. C , D >. e. U_ x e. A ( { x } X. B ) )
2		relxp 4608	. . . . . 6 |- Rel ( { x } X. B )
3	2	rgenw 2461	. . . . 5 |- A. x e. A Rel ( { x } X. B )
4		reliun 4620	. . . . 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 4542	. . 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 2668	. . 3 |- ( C e. A -> C e. _V )
9	8	adantr 272	. 2 |- ( ( C e. A /\ D e. E ) -> C e. _V )
10		nfcv 2255	. . 3 |- F/_ x C
11		nfiu1 3809	. . . . 5 |- F/_ x U_ x e. A ( { x } X. B )
12	11	nfel2 2268	. . . 4 |- F/ x <. C , D >. e. U_ x e. A ( { x } X. B )
13		nfv 1491	. . . 4 |- F/ x ( C e. A /\ D e. E )
14	12, 13	nfbi 1551	. . 3 |- F/ x ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )
15		opeq1 3671	. . . . 5 |- ( x = C -> <. x , D >. = <. C , D >. )
16	15	eleq1d 2183	. . . 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 2177	. . . . 5 |- ( x = C -> ( x e. A <-> C e. A ) )
18		opeliunxp2.1	. . . . . 6 |- ( x = C -> B = E )
19	18	eleq2d 2184	. . . . 5 |- ( x = C -> ( D e. B <-> D e. E ) )
20	17, 19	anbi12d 462	. . . 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 4554	. . 3 |- ( <. x , D >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ D e. B ) )
23	10, 14, 21, 22	vtoclgf 2715	. 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 676	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 1314   e. wcel 1463   A.wral 2390   _Vcvv 2657   {csn 3493   <.cop 3496   U_ciun 3779   class class class wbr 3895   X. cxp 4497   Rel wrel 4504

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-sbc 2879  df-csb 2972  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-iun 3781  df-br 3896  df-opab 3950  df-xp 4505  df-rel 4506

This theorem is referenced by:  mpoxopn0yelv  6090  eldvap  12606