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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
StepHypRef 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 )
32rgenw 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
) )
53, 4mpbir 146 . . . 4  |-  Rel  U_ x  e.  A  ( {
x }  X.  B
)
65brrelex1i 4717 . . 3  |-  ( C
U_ x  e.  A  ( { x }  X.  B ) D  ->  C  e.  _V )
71, 6sylbir 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 )
98adantr 276 . 2  |-  ( ( C  e.  A  /\  D  e.  E )  ->  C  e.  _V )
10 nfiu1 3956 . . . . 5  |-  F/_ x U_ x  e.  A  ( { x }  X.  B )
1110nfel2 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
1413nfel2 2360 . . . . 5  |-  F/ x  D  e.  E
1512, 14nfan 1587 . . . 4  |-  F/ x
( C  e.  A  /\  D  e.  E
)
1611, 15nfbi 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 >. )
1817eleq1d 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 )
2120eleq2d 2274 . . . . 5  |-  ( x  =  C  ->  ( D  e.  B  <->  D  e.  E ) )
2219, 21anbi12d 473 . . . 4  |-  ( x  =  C  ->  (
( x  e.  A  /\  D  e.  B
)  <->  ( C  e.  A  /\  D  e.  E ) ) )
2318, 22bibi12d 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 ) )
2516, 23, 24vtoclg1f 2831 . 2  |-  ( C  e.  _V  ->  ( <. C ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  ( C  e.  A  /\  D  e.  E ) ) )
267, 9, 25pm5.21nii 705 1  |-  ( <. C ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  ( C  e.  A  /\  D  e.  E ) )
Colors of variables: wff set class
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
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