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Theorem opeliunxp2 4649
Description: Membership in a union of cross products. (Contributed by Mario Carneiro, 14-Feb-2015.)
Hypothesis
Ref Expression
opeliunxp2.1  |-  ( x  =  C  ->  B  =  E )
Assertion
Ref Expression
opeliunxp2  |-  ( <. C ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  ( C  e.  A  /\  D  e.  E ) )
Distinct variable groups:    x, C    x, D    x, E    x, A
Allowed substitution hint:    B( x)

Proof of Theorem opeliunxp2
StepHypRef Expression
1 df-br 3900 . . 3  |-  ( C
U_ x  e.  A  ( { x }  X.  B ) D  <->  <. C ,  D >.  e.  U_ x  e.  A  ( {
x }  X.  B
) )
2 relxp 4618 . . . . . 6  |-  Rel  ( { x }  X.  B )
32rgenw 2464 . . . . 5  |-  A. x  e.  A  Rel  ( { x }  X.  B
)
4 reliun 4630 . . . . 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 4552 . . 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 2671 . . 3  |-  ( C  e.  A  ->  C  e.  _V )
98adantr 274 . 2  |-  ( ( C  e.  A  /\  D  e.  E )  ->  C  e.  _V )
10 nfcv 2258 . . 3  |-  F/_ x C
11 nfiu1 3813 . . . . 5  |-  F/_ x U_ x  e.  A  ( { x }  X.  B )
1211nfel2 2271 . . . 4  |-  F/ x <. C ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )
13 nfv 1493 . . . 4  |-  F/ x
( C  e.  A  /\  D  e.  E
)
1412, 13nfbi 1553 . . 3  |-  F/ x
( <. C ,  D >.  e.  U_ x  e.  A  ( { x }  X.  B )  <->  ( C  e.  A  /\  D  e.  E ) )
15 opeq1 3675 . . . . 5  |-  ( x  =  C  ->  <. x ,  D >.  =  <. C ,  D >. )
1615eleq1d 2186 . . . 4  |-  ( x  =  C  ->  ( <. x ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  <. C ,  D >.  e.  U_ x  e.  A  ( {
x }  X.  B
) ) )
17 eleq1 2180 . . . . 5  |-  ( x  =  C  ->  (
x  e.  A  <->  C  e.  A ) )
18 opeliunxp2.1 . . . . . 6  |-  ( x  =  C  ->  B  =  E )
1918eleq2d 2187 . . . . 5  |-  ( x  =  C  ->  ( D  e.  B  <->  D  e.  E ) )
2017, 19anbi12d 464 . . . 4  |-  ( x  =  C  ->  (
( x  e.  A  /\  D  e.  B
)  <->  ( C  e.  A  /\  D  e.  E ) ) )
2116, 20bibi12d 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 ) ) ) )
22 opeliunxp 4564 . . 3  |-  ( <.
x ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  ( x  e.  A  /\  D  e.  B ) )
2310, 14, 21, 22vtoclgf 2718 . 2  |-  ( C  e.  _V  ->  ( <. C ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  ( C  e.  A  /\  D  e.  E ) ) )
247, 9, 23pm5.21nii 678 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 1316    e. wcel 1465   A.wral 2393   _Vcvv 2660   {csn 3497   <.cop 3500   U_ciun 3783   class class class wbr 3899    X. cxp 4507   Rel wrel 4514
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-iun 3785  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516
This theorem is referenced by:  mpoxopn0yelv  6104  eldvap  12731
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