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Theorem opeliunxp2f 6003
Description: Membership in a union of Cartesian products, using bound-variable hypothesis for  E instead of distinct variable conditions as in opeliunxp2 4576. (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 3846 . . 3  |-  ( C
U_ x  e.  A  ( { x }  X.  B ) D  <->  <. C ,  D >.  e.  U_ x  e.  A  ( {
x }  X.  B
) )
2 relxp 4547 . . . . . 6  |-  Rel  ( { x }  X.  B )
32rgenw 2430 . . . . 5  |-  A. x  e.  A  Rel  ( { x }  X.  B
)
4 reliun 4558 . . . . 5  |-  ( Rel  U_ x  e.  A  ( { x }  X.  B )  <->  A. x  e.  A  Rel  ( { x }  X.  B
) )
53, 4mpbir 144 . . . 4  |-  Rel  U_ x  e.  A  ( {
x }  X.  B
)
65brrelex1i 4481 . . 3  |-  ( C
U_ x  e.  A  ( { x }  X.  B ) D  ->  C  e.  _V )
71, 6sylbir 133 . 2  |-  ( <. C ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )  ->  C  e.  _V )
8 elex 2630 . . 3  |-  ( C  e.  A  ->  C  e.  _V )
98adantr 270 . 2  |-  ( ( C  e.  A  /\  D  e.  E )  ->  C  e.  _V )
10 nfiu1 3760 . . . . 5  |-  F/_ x U_ x  e.  A  ( { x }  X.  B )
1110nfel2 2241 . . . 4  |-  F/ x <. C ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )
12 nfv 1466 . . . . 5  |-  F/ x  C  e.  A
13 opeliunxp2f.f . . . . . 6  |-  F/_ x E
1413nfel2 2241 . . . . 5  |-  F/ x  D  e.  E
1512, 14nfan 1502 . . . 4  |-  F/ x
( C  e.  A  /\  D  e.  E
)
1611, 15nfbi 1526 . . 3  |-  F/ x
( <. C ,  D >.  e.  U_ x  e.  A  ( { x }  X.  B )  <->  ( C  e.  A  /\  D  e.  E ) )
17 opeq1 3622 . . . . 5  |-  ( x  =  C  ->  <. x ,  D >.  =  <. C ,  D >. )
1817eleq1d 2156 . . . 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 2150 . . . . 5  |-  ( x  =  C  ->  (
x  e.  A  <->  C  e.  A ) )
20 opeliunxp2f.e . . . . . 6  |-  ( x  =  C  ->  B  =  E )
2120eleq2d 2157 . . . . 5  |-  ( x  =  C  ->  ( D  e.  B  <->  D  e.  E ) )
2219, 21anbi12d 457 . . . 4  |-  ( x  =  C  ->  (
( x  e.  A  /\  D  e.  B
)  <->  ( C  e.  A  /\  D  e.  E ) ) )
2318, 22bibi12d 233 . . 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 4493 . . 3  |-  ( <.
x ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  ( x  e.  A  /\  D  e.  B ) )
2516, 23, 24vtoclg1f 2678 . 2  |-  ( C  e.  _V  ->  ( <. C ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  ( C  e.  A  /\  D  e.  E ) ) )
267, 9, 25pm5.21nii 655 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 102    <-> wb 103    = wceq 1289    e. wcel 1438   F/_wnfc 2215   A.wral 2359   _Vcvv 2619   {csn 3446   <.cop 3449   U_ciun 3730   class class class wbr 3845    X. cxp 4436   Rel wrel 4443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-iun 3732  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445
This theorem is referenced by:  fisumcom2  10832
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