Theorem opeliunxp2 4769

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 4006	. . 3 |- ( C U_ x e. A ( { x } X. B ) D <-> <. C , D >. e. U_ x e. A ( { x } X. B ) )
2		relxp 4737	. . . . . 6 |- Rel ( { x } X. B )
3	2	rgenw 2532	. . . . 5 |- A. x e. A Rel ( { x } X. B )
4		reliun 4749	. . . . 5 |- ( Rel U_ x e. A ( { x } X. B ) <-> A. x e. A Rel ( { x } X. B ) )
5	3, 4	mpbir 146	. . . 4 |- Rel U_ x e. A ( { x } X. B )
6	5	brrelex1i 4671	. . 3 |- ( C U_ x e. A ( { x } X. B ) D -> C e. _V )
7	1, 6	sylbir 135	. 2 |- ( <. C , D >. e. U_ x e. A ( { x } X. B ) -> C e. _V )
8		elex 2750	. . 3 |- ( C e. A -> C e. _V )
9	8	adantr 276	. 2 |- ( ( C e. A /\ D e. E ) -> C e. _V )
10		nfcv 2319	. . 3 |- F/_ x C
11		nfiu1 3918	. . . . 5 |- F/_ x U_ x e. A ( { x } X. B )
12	11	nfel2 2332	. . . 4 |- F/ x <. C , D >. e. U_ x e. A ( { x } X. B )
13		nfv 1528	. . . 4 |- F/ x ( C e. A /\ D e. E )
14	12, 13	nfbi 1589	. . 3 |- F/ x ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )
15		opeq1 3780	. . . . 5 |- ( x = C -> <. x , D >. = <. C , D >. )
16	15	eleq1d 2246	. . . 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 2240	. . . . 5 |- ( x = C -> ( x e. A <-> C e. A ) )
18		opeliunxp2.1	. . . . . 6 |- ( x = C -> B = E )
19	18	eleq2d 2247	. . . . 5 |- ( x = C -> ( D e. B <-> D e. E ) )
20	17, 19	anbi12d 473	. . . 4 |- ( x = C -> ( ( x e. A /\ D e. B ) <-> ( C e. A /\ D e. E ) ) )
21	16, 20	bibi12d 235	. . 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 4683	. . 3 |- ( <. x , D >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ D e. B ) )
23	10, 14, 21, 22	vtoclgf 2797	. 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 704	1 |- ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   A.wral 2455   _Vcvv 2739   {csn 3594   <.cop 3597   U_ciun 3888   class class class wbr 4005   X. cxp 4626   Rel wrel 4633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-iun 3890  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635

This theorem is referenced by:  mpoxopn0yelv  6242  eldvap  14236