Theorem opeliunxp2f 6469

Description: Membership in a union of Cartesian products, using bound-variable hypothesis for E instead of distinct variable conditions as in opeliunxp2 4895. (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 4110	. . 3 |- ( C U_ x e. A ( { x } X. B ) D <-> <. C , D >. e. U_ x e. A ( { x } X. B ) )
2		relxp 4859	. . . . . 6 |- Rel ( { x } X. B )
3	2	rgenw 2597	. . . . 5 |- A. x e. A Rel ( { x } X. B )
4		reliun 4873	. . . . 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 4793	. . 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 2825	. . 3 |- ( C e. A -> C e. _V )
9	8	adantr 276	. 2 |- ( ( C e. A /\ D e. E ) -> C e. _V )
10		nfiu1 4021	. . . . 5 |- F/_ x U_ x e. A ( { x } X. B )
11	10	nfel2 2397	. . . 4 |- F/ x <. C , D >. e. U_ x e. A ( { x } X. B )
12		nfv 1577	. . . . 5 |- F/ x C e. A
13		opeliunxp2f.f	. . . . . 6 |- F/_ x E
14	13	nfel2 2397	. . . . 5 |- F/ x D e. E
15	12, 14	nfan 1614	. . . 4 |- F/ x ( C e. A /\ D e. E )
16	11, 15	nfbi 1638	. . 3 |- F/ x ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )
17		opeq1 3883	. . . . 5 |- ( x = C -> <. x , D >. = <. C , D >. )
18	17	eleq1d 2301	. . . 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 2295	. . . . 5 |- ( x = C -> ( x e. A <-> C e. A ) )
20		opeliunxp2f.e	. . . . . 6 |- ( x = C -> B = E )
21	20	eleq2d 2302	. . . . 5 |- ( x = C -> ( D e. B <-> D e. E ) )
22	19, 21	anbi12d 473	. . . 4 |- ( x = C -> ( ( x e. A /\ D e. B ) <-> ( C e. A /\ D e. E ) ) )
23	18, 22	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 ) ) ) )
24		opeliunxp 4805	. . 3 |- ( <. x , D >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ D e. B ) )
25	16, 23, 24	vtoclg1f 2874	. 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 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 2203   F/_wnfc 2371   A.wral 2520   _Vcvv 2813   {csn 3689   <.cop 3692   U_ciun 3991   class class class wbr 4109   X. cxp 4747   Rel wrel 4754

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-iun 3993  df-br 4110  df-opab 4172  df-xp 4755  df-rel 4756

This theorem is referenced by:  fisumcom2  12124  fprodcom2fi  12312