Theorem opeliunxp2 4876

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 4094	. . 3 |- ( C U_ x e. A ( { x } X. B ) D <-> <. C , D >. e. U_ x e. A ( { x } X. B ) )
2		relxp 4841	. . . . . 6 |- Rel ( { x } X. B )
3	2	rgenw 2588	. . . . 5 |- A. x e. A Rel ( { x } X. B )
4		reliun 4854	. . . . 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 4775	. . 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 2815	. . 3 |- ( C e. A -> C e. _V )
9	8	adantr 276	. 2 |- ( ( C e. A /\ D e. E ) -> C e. _V )
10		nfcv 2375	. . 3 |- F/_ x C
11		nfiu1 4005	. . . . 5 |- F/_ x U_ x e. A ( { x } X. B )
12	11	nfel2 2388	. . . 4 |- F/ x <. C , D >. e. U_ x e. A ( { x } X. B )
13		nfv 1577	. . . 4 |- F/ x ( C e. A /\ D e. E )
14	12, 13	nfbi 1638	. . 3 |- F/ x ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )
15		opeq1 3867	. . . . 5 |- ( x = C -> <. x , D >. = <. C , D >. )
16	15	eleq1d 2300	. . . 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 2294	. . . . 5 |- ( x = C -> ( x e. A <-> C e. A ) )
18		opeliunxp2.1	. . . . . 6 |- ( x = C -> B = E )
19	18	eleq2d 2301	. . . . 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 4787	. . 3 |- ( <. x , D >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ D e. B ) )
23	10, 14, 21, 22	vtoclgf 2863	. 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 712	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 1398   e. wcel 2202   A.wral 2511   _Vcvv 2803   {csn 3673   <.cop 3676   U_ciun 3975   class class class wbr 4093   X. cxp 4729   Rel wrel 4736

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-iun 3977  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738

This theorem is referenced by:  mpoxopn0yelv  6448  eldvap  15493