Theorem opeliunxp2f 6323

Description: Membership in a union of Cartesian products, using bound-variable hypothesis for E instead of distinct variable conditions as in opeliunxp2 4817. (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 4044	. . 3 |- ( C U_ x e. A ( { x } X. B ) D <-> <. C , D >. e. U_ x e. A ( { x } X. B ) )
2		relxp 4783	. . . . . 6 |- Rel ( { x } X. B )
3	2	rgenw 2560	. . . . 5 |- A. x e. A Rel ( { x } X. B )
4		reliun 4795	. . . . 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 4717	. . 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 2782	. . 3 |- ( C e. A -> C e. _V )
9	8	adantr 276	. 2 |- ( ( C e. A /\ D e. E ) -> C e. _V )
10		nfiu1 3956	. . . . 5 |- F/_ x U_ x e. A ( { x } X. B )
11	10	nfel2 2360	. . . 4 |- F/ x <. C , D >. e. U_ x e. A ( { x } X. B )
12		nfv 1550	. . . . 5 |- F/ x C e. A
13		opeliunxp2f.f	. . . . . 6 |- F/_ x E
14	13	nfel2 2360	. . . . 5 |- F/ x D e. E
15	12, 14	nfan 1587	. . . 4 |- F/ x ( C e. A /\ D e. E )
16	11, 15	nfbi 1611	. . 3 |- F/ x ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )
17		opeq1 3818	. . . . 5 |- ( x = C -> <. x , D >. = <. C , D >. )
18	17	eleq1d 2273	. . . 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 2267	. . . . 5 |- ( x = C -> ( x e. A <-> C e. A ) )
20		opeliunxp2f.e	. . . . . 6 |- ( x = C -> B = E )
21	20	eleq2d 2274	. . . . 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 4729	. . 3 |- ( <. x , D >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ D e. B ) )
25	16, 23, 24	vtoclg1f 2831	. 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 705	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 1372   e. wcel 2175   F/_wnfc 2334   A.wral 2483   _Vcvv 2771   {csn 3632   <.cop 3635   U_ciun 3926   class class class wbr 4043   X. cxp 4672   Rel wrel 4679

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-iun 3928  df-br 4044  df-opab 4105  df-xp 4680  df-rel 4681

This theorem is referenced by:  fisumcom2  11720  fprodcom2fi  11908