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Theorem opeliunxp2f 6982
Description: Membership in a union of Cartesian products, using bound-variable hypothesis for  E instead of distinct variable conditions as in opeliunxp2 4991. (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
StepHypRef Expression
1 df-br 4416 . . 3  |-  ( C
U_ x  e.  A  ( { x }  X.  B ) D  <->  <. C ,  D >.  e.  U_ x  e.  A  ( {
x }  X.  B
) )
2 relxp 4960 . . . . . 6  |-  Rel  ( { x }  X.  B )
32rgenw 2760 . . . . 5  |-  A. x  e.  A  Rel  ( { x }  X.  B
)
4 reliun 4972 . . . . 5  |-  ( Rel  U_ x  e.  A  ( { x }  X.  B )  <->  A. x  e.  A  Rel  ( { x }  X.  B
) )
53, 4mpbir 214 . . . 4  |-  Rel  U_ x  e.  A  ( {
x }  X.  B
)
65brrelexi 4893 . . 3  |-  ( C
U_ x  e.  A  ( { x }  X.  B ) D  ->  C  e.  _V )
71, 6sylbir 218 . 2  |-  ( <. C ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )  ->  C  e.  _V )
8 elex 3065 . . 3  |-  ( C  e.  A  ->  C  e.  _V )
98adantr 471 . 2  |-  ( ( C  e.  A  /\  D  e.  E )  ->  C  e.  _V )
10 nfiu1 4321 . . . . 5  |-  F/_ x U_ x  e.  A  ( { x }  X.  B )
1110nfel2 2618 . . . 4  |-  F/ x <. C ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )
12 nfv 1771 . . . . 5  |-  F/ x  C  e.  A
13 opeliunxp2f.f . . . . . 6  |-  F/_ x E
1413nfel2 2618 . . . . 5  |-  F/ x  D  e.  E
1512, 14nfan 2021 . . . 4  |-  F/ x
( C  e.  A  /\  D  e.  E
)
1611, 15nfbi 2027 . . 3  |-  F/ x
( <. C ,  D >.  e.  U_ x  e.  A  ( { x }  X.  B )  <->  ( C  e.  A  /\  D  e.  E ) )
17 opeq1 4179 . . . . 5  |-  ( x  =  C  ->  <. x ,  D >.  =  <. C ,  D >. )
1817eleq1d 2523 . . . 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 2527 . . . . 5  |-  ( x  =  C  ->  (
x  e.  A  <->  C  e.  A ) )
20 opeliunxp2f.e . . . . . 6  |-  ( x  =  C  ->  B  =  E )
2120eleq2d 2524 . . . . 5  |-  ( x  =  C  ->  ( D  e.  B  <->  D  e.  E ) )
2219, 21anbi12d 722 . . . 4  |-  ( x  =  C  ->  (
( x  e.  A  /\  D  e.  B
)  <->  ( C  e.  A  /\  D  e.  E ) ) )
2318, 22bibi12d 327 . . 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 4904 . . 3  |-  ( <.
x ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  ( x  e.  A  /\  D  e.  B ) )
2516, 23, 24vtoclg1f 3117 . 2  |-  ( C  e.  _V  ->  ( <. C ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  ( C  e.  A  /\  D  e.  E ) ) )
267, 9, 25pm5.21nii 359 1  |-  ( <. C ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  ( C  e.  A  /\  D  e.  E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1454    e. wcel 1897   F/_wnfc 2589   A.wral 2748   _Vcvv 3056   {csn 3979   <.cop 3985   U_ciun 4291   class class class wbr 4415    X. cxp 4850   Rel wrel 4857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-iun 4293  df-br 4416  df-opab 4475  df-xp 4858  df-rel 4859
This theorem is referenced by:  mpt2xeldm  6983
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