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Theorem opeliunxp2f 6135
Description: Membership in a union of Cartesian products, using bound-variable hypothesis for  E instead of distinct variable conditions as in opeliunxp2 4679. (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 3930 . . 3  |-  ( C
U_ x  e.  A  ( { x }  X.  B ) D  <->  <. C ,  D >.  e.  U_ x  e.  A  ( {
x }  X.  B
) )
2 relxp 4648 . . . . . 6  |-  Rel  ( { x }  X.  B )
32rgenw 2487 . . . . 5  |-  A. x  e.  A  Rel  ( { x }  X.  B
)
4 reliun 4660 . . . . 5  |-  ( Rel  U_ x  e.  A  ( { x }  X.  B )  <->  A. x  e.  A  Rel  ( { x }  X.  B
) )
53, 4mpbir 145 . . . 4  |-  Rel  U_ x  e.  A  ( {
x }  X.  B
)
65brrelex1i 4582 . . 3  |-  ( C
U_ x  e.  A  ( { x }  X.  B ) D  ->  C  e.  _V )
71, 6sylbir 134 . 2  |-  ( <. C ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )  ->  C  e.  _V )
8 elex 2697 . . 3  |-  ( C  e.  A  ->  C  e.  _V )
98adantr 274 . 2  |-  ( ( C  e.  A  /\  D  e.  E )  ->  C  e.  _V )
10 nfiu1 3843 . . . . 5  |-  F/_ x U_ x  e.  A  ( { x }  X.  B )
1110nfel2 2294 . . . 4  |-  F/ x <. C ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )
12 nfv 1508 . . . . 5  |-  F/ x  C  e.  A
13 opeliunxp2f.f . . . . . 6  |-  F/_ x E
1413nfel2 2294 . . . . 5  |-  F/ x  D  e.  E
1512, 14nfan 1544 . . . 4  |-  F/ x
( C  e.  A  /\  D  e.  E
)
1611, 15nfbi 1568 . . 3  |-  F/ x
( <. C ,  D >.  e.  U_ x  e.  A  ( { x }  X.  B )  <->  ( C  e.  A  /\  D  e.  E ) )
17 opeq1 3705 . . . . 5  |-  ( x  =  C  ->  <. x ,  D >.  =  <. C ,  D >. )
1817eleq1d 2208 . . . 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 2202 . . . . 5  |-  ( x  =  C  ->  (
x  e.  A  <->  C  e.  A ) )
20 opeliunxp2f.e . . . . . 6  |-  ( x  =  C  ->  B  =  E )
2120eleq2d 2209 . . . . 5  |-  ( x  =  C  ->  ( D  e.  B  <->  D  e.  E ) )
2219, 21anbi12d 464 . . . 4  |-  ( x  =  C  ->  (
( x  e.  A  /\  D  e.  B
)  <->  ( C  e.  A  /\  D  e.  E ) ) )
2318, 22bibi12d 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 4594 . . 3  |-  ( <.
x ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  ( x  e.  A  /\  D  e.  B ) )
2516, 23, 24vtoclg1f 2745 . 2  |-  ( C  e.  _V  ->  ( <. C ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  ( C  e.  A  /\  D  e.  E ) ) )
267, 9, 25pm5.21nii 693 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 103    <-> wb 104    = wceq 1331    e. wcel 1480   F/_wnfc 2268   A.wral 2416   _Vcvv 2686   {csn 3527   <.cop 3530   U_ciun 3813   class class class wbr 3929    X. cxp 4537   Rel wrel 4544
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-iun 3815  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546
This theorem is referenced by:  fisumcom2  11219
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