Theorem opeliunxp2f 6206

Description: Membership in a union of Cartesian products, using bound-variable hypothesis for E instead of distinct variable conditions as in opeliunxp2 4744. (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 3983	. . 3 |- ( C U_ x e. A ( { x } X. B ) D <-> <. C , D >. e. U_ x e. A ( { x } X. B ) )
2		relxp 4713	. . . . . 6 |- Rel ( { x } X. B )
3	2	rgenw 2521	. . . . 5 |- A. x e. A Rel ( { x } X. B )
4		reliun 4725	. . . . 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 4647	. . 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 2737	. . 3 |- ( C e. A -> C e. _V )
9	8	adantr 274	. 2 |- ( ( C e. A /\ D e. E ) -> C e. _V )
10		nfiu1 3896	. . . . 5 |- F/_ x U_ x e. A ( { x } X. B )
11	10	nfel2 2321	. . . 4 |- F/ x <. C , D >. e. U_ x e. A ( { x } X. B )
12		nfv 1516	. . . . 5 |- F/ x C e. A
13		opeliunxp2f.f	. . . . . 6 |- F/_ x E
14	13	nfel2 2321	. . . . 5 |- F/ x D e. E
15	12, 14	nfan 1553	. . . 4 |- F/ x ( C e. A /\ D e. E )
16	11, 15	nfbi 1577	. . 3 |- F/ x ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )
17		opeq1 3758	. . . . 5 |- ( x = C -> <. x , D >. = <. C , D >. )
18	17	eleq1d 2235	. . . 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 2229	. . . . 5 |- ( x = C -> ( x e. A <-> C e. A ) )
20		opeliunxp2f.e	. . . . . 6 |- ( x = C -> B = E )
21	20	eleq2d 2236	. . . . 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 4659	. . 3 |- ( <. x , D >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ D e. B ) )
25	16, 23, 24	vtoclg1f 2785	. 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 1343   e. wcel 2136   F/_wnfc 2295   A.wral 2444   _Vcvv 2726   {csn 3576   <.cop 3579   U_ciun 3866   class class class wbr 3982   X. cxp 4602   Rel wrel 4609

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-iun 3868  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611

This theorem is referenced by:  fisumcom2  11379  fprodcom2fi  11567