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Theorem opeliunxp2 4840
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 4040 . . 3  |-  ( C
U_ x  e.  A  ( { x }  X.  B ) D  <->  <. C ,  D >.  e.  U_ x  e.  A  ( {
x }  X.  B
) )
2 relxp 4810 . . . . . 6  |-  Rel  ( { x }  X.  B )
32rgenw 2623 . . . . 5  |-  A. x  e.  A  Rel  ( { x }  X.  B
)
4 reliun 4822 . . . . 5  |-  ( Rel  U_ x  e.  A  ( { x }  X.  B )  <->  A. x  e.  A  Rel  ( { x }  X.  B
) )
53, 4mpbir 200 . . . 4  |-  Rel  U_ x  e.  A  ( {
x }  X.  B
)
65brrelexi 4745 . . 3  |-  ( C
U_ x  e.  A  ( { x }  X.  B ) D  ->  C  e.  _V )
71, 6sylbir 204 . 2  |-  ( <. C ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )  ->  C  e.  _V )
8 elex 2809 . . 3  |-  ( C  e.  A  ->  C  e.  _V )
98adantr 451 . 2  |-  ( ( C  e.  A  /\  D  e.  E )  ->  C  e.  _V )
10 nfcv 2432 . . 3  |-  F/_ x C
11 nfiu1 3949 . . . . 5  |-  F/_ x U_ x  e.  A  ( { x }  X.  B )
1211nfel2 2444 . . . 4  |-  F/ x <. C ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )
13 nfv 1609 . . . 4  |-  F/ x
( C  e.  A  /\  D  e.  E
)
1412, 13nfbi 1784 . . 3  |-  F/ x
( <. C ,  D >.  e.  U_ x  e.  A  ( { x }  X.  B )  <->  ( C  e.  A  /\  D  e.  E ) )
15 opeq1 3812 . . . . 5  |-  ( x  =  C  ->  <. x ,  D >.  =  <. C ,  D >. )
1615eleq1d 2362 . . . 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 2356 . . . . 5  |-  ( x  =  C  ->  (
x  e.  A  <->  C  e.  A ) )
18 opeliunxp2.1 . . . . . 6  |-  ( x  =  C  ->  B  =  E )
1918eleq2d 2363 . . . . 5  |-  ( x  =  C  ->  ( D  e.  B  <->  D  e.  E ) )
2017, 19anbi12d 691 . . . 4  |-  ( x  =  C  ->  (
( x  e.  A  /\  D  e.  B
)  <->  ( C  e.  A  /\  D  e.  E ) ) )
2116, 20bibi12d 312 . . 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 4756 . . 3  |-  ( <.
x ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  ( x  e.  A  /\  D  e.  B ) )
2310, 14, 21, 22vtoclgf 2855 . 2  |-  ( C  e.  _V  ->  ( <. C ,  D >.  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  ( C  e.  A  /\  D  e.  E ) ) )
247, 9, 23pm5.21nii 342 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    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801   {csn 3653   <.cop 3656   U_ciun 3921   class class class wbr 4039    X. cxp 4703   Rel wrel 4710
This theorem is referenced by:  eldmcoa  13913  dmdprd  15252  eldv  19264  perfdvf  19269  eltayl  19755  cvmliftlem1  23831  filnetlem3  26432  mpt2xopn0yelv  28195  mpt2xopxnop0  28197  nbgrael  28275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-iun 3923  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712
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