Theorem opeliunxp2f 6143

Description: Membership in a union of Cartesian products, using bound-variable hypothesis for E instead of distinct variable conditions as in opeliunxp2 4687. (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 3938	. . 3 |- ( C U_ x e. A ( { x } X. B ) D <-> <. C , D >. e. U_ x e. A ( { x } X. B ) )
2		relxp 4656	. . . . . 6 |- Rel ( { x } X. B )
3	2	rgenw 2490	. . . . 5 |- A. x e. A Rel ( { x } X. B )
4		reliun 4668	. . . . 5 |- ( Rel U_ x e. A ( { x } X. B ) <-> A. x e. A Rel ( { x } X. B ) )
5	3, 4	mpbir 145	. . . 4 |- Rel U_ x e. A ( { x } X. B )
6	5	brrelex1i 4590	. . 3 |- ( C U_ x e. A ( { x } X. B ) D -> C e. _V )
7	1, 6	sylbir 134	. 2 |- ( <. C , D >. e. U_ x e. A ( { x } X. B ) -> C e. _V )
8		elex 2700	. . 3 |- ( C e. A -> C e. _V )
9	8	adantr 274	. 2 |- ( ( C e. A /\ D e. E ) -> C e. _V )
10		nfiu1 3851	. . . . 5 |- F/_ x U_ x e. A ( { x } X. B )
11	10	nfel2 2295	. . . 4 |- F/ x <. C , D >. e. U_ x e. A ( { x } X. B )
12		nfv 1509	. . . . 5 |- F/ x C e. A
13		opeliunxp2f.f	. . . . . 6 |- F/_ x E
14	13	nfel2 2295	. . . . 5 |- F/ x D e. E
15	12, 14	nfan 1545	. . . 4 |- F/ x ( C e. A /\ D e. E )
16	11, 15	nfbi 1569	. . 3 |- F/ x ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )
17		opeq1 3713	. . . . 5 |- ( x = C -> <. x , D >. = <. C , D >. )
18	17	eleq1d 2209	. . . 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 2203	. . . . 5 |- ( x = C -> ( x e. A <-> C e. A ) )
20		opeliunxp2f.e	. . . . . 6 |- ( x = C -> B = E )
21	20	eleq2d 2210	. . . . 5 |- ( x = C -> ( D e. B <-> D e. E ) )
22	19, 21	anbi12d 465	. . . 4 |- ( x = C -> ( ( x e. A /\ D e. B ) <-> ( C e. A /\ D e. E ) ) )
23	18, 22	bibi12d 234	. . 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 4602	. . 3 |- ( <. x , D >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ D e. B ) )
25	16, 23, 24	vtoclg1f 2748	. 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 694	1 |- ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   F/_wnfc 2269   A.wral 2417   _Vcvv 2689   {csn 3532   <.cop 3535   U_ciun 3821   class class class wbr 3937   X. cxp 4545   Rel wrel 4552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-iun 3823  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554

This theorem is referenced by:  fisumcom2  11239