Theorem opeliunxp2f 6003

Description: Membership in a union of Cartesian products, using bound-variable hypothesis for E instead of distinct variable conditions as in opeliunxp2 4576. (Contributed by AV, 25-Oct-2020.)

Hypotheses

Ref	Expression
opeliunxp2f.f	|- F/_ x E
opeliunxp2f.e	|- ( x = C -> B = E )

Assertion

Ref	Expression
opeliunxp2f	|- ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )

Distinct variable groups:   x, A   x, C   x, D

Allowed substitution hints:   B( x)   E( x)

Proof of Theorem opeliunxp2f

Step	Hyp	Ref	Expression
1		df-br 3846	. . 3 |- ( C U_ x e. A ( { x } X. B ) D <-> <. C , D >. e. U_ x e. A ( { x } X. B ) )
2		relxp 4547	. . . . . 6 |- Rel ( { x } X. B )
3	2	rgenw 2430	. . . . 5 |- A. x e. A Rel ( { x } X. B )
4		reliun 4558	. . . . 5 |- ( Rel U_ x e. A ( { x } X. B ) <-> A. x e. A Rel ( { x } X. B ) )
5	3, 4	mpbir 144	. . . 4 |- Rel U_ x e. A ( { x } X. B )
6	5	brrelex1i 4481	. . 3 |- ( C U_ x e. A ( { x } X. B ) D -> C e. _V )
7	1, 6	sylbir 133	. 2 |- ( <. C , D >. e. U_ x e. A ( { x } X. B ) -> C e. _V )
8		elex 2630	. . 3 |- ( C e. A -> C e. _V )
9	8	adantr 270	. 2 |- ( ( C e. A /\ D e. E ) -> C e. _V )
10		nfiu1 3760	. . . . 5 |- F/_ x U_ x e. A ( { x } X. B )
11	10	nfel2 2241	. . . 4 |- F/ x <. C , D >. e. U_ x e. A ( { x } X. B )
12		nfv 1466	. . . . 5 |- F/ x C e. A
13		opeliunxp2f.f	. . . . . 6 |- F/_ x E
14	13	nfel2 2241	. . . . 5 |- F/ x D e. E
15	12, 14	nfan 1502	. . . 4 |- F/ x ( C e. A /\ D e. E )
16	11, 15	nfbi 1526	. . 3 |- F/ x ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )
17		opeq1 3622	. . . . 5 |- ( x = C -> <. x , D >. = <. C , D >. )
18	17	eleq1d 2156	. . . 4 |- ( x = C -> ( <. x , D >. e. U_ x e. A ( { x } X. B ) <-> <. C , D >. e. U_ x e. A ( { x } X. B ) ) )
19		eleq1 2150	. . . . 5 |- ( x = C -> ( x e. A <-> C e. A ) )
20		opeliunxp2f.e	. . . . . 6 |- ( x = C -> B = E )
21	20	eleq2d 2157	. . . . 5 |- ( x = C -> ( D e. B <-> D e. E ) )
22	19, 21	anbi12d 457	. . . 4 |- ( x = C -> ( ( x e. A /\ D e. B ) <-> ( C e. A /\ D e. E ) ) )
23	18, 22	bibi12d 233	. . 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 ) ) ) )
24		opeliunxp 4493	. . 3 |- ( <. x , D >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ D e. B ) )
25	16, 23, 24	vtoclg1f 2678	. 2 |- ( C e. _V -> ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) ) )
26	7, 9, 25	pm5.21nii 655	1 |- ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   e. wcel 1438   F/_wnfc 2215   A.wral 2359   _Vcvv 2619   {csn 3446   <.cop 3449   U_ciun 3730   class class class wbr 3845   X. cxp 4436   Rel wrel 4443

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-iun 3732  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445

This theorem is referenced by:  fisumcom2  10832