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Theorem eliunxp 4823
Description: Membership in a union of cross products. Analogue of elxp 4706 for nonconstant  B ( x ). (Contributed by Mario Carneiro, 29-Dec-2014.)
Assertion
Ref Expression
eliunxp  |-  ( C  e.  U_ x  e.  A  ( { x }  X.  B )  <->  E. x E. y ( C  = 
<. x ,  y >.  /\  ( x  e.  A  /\  y  e.  B
) ) )
Distinct variable groups:    y, A    y, B    x, y, C
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem eliunxp
StepHypRef Expression
1 relxp 4794 . . . . . 6  |-  Rel  ( { x }  X.  B )
21rgenw 2610 . . . . 5  |-  A. x  e.  A  Rel  ( { x }  X.  B
)
3 reliun 4806 . . . . 5  |-  ( Rel  U_ x  e.  A  ( { x }  X.  B )  <->  A. x  e.  A  Rel  ( { x }  X.  B
) )
42, 3mpbir 200 . . . 4  |-  Rel  U_ x  e.  A  ( {
x }  X.  B
)
5 elrel 4789 . . . 4  |-  ( ( Rel  U_ x  e.  A  ( { x }  X.  B )  /\  C  e.  U_ x  e.  A  ( { x }  X.  B ) )  ->  E. x E. y  C  =  <. x ,  y
>. )
64, 5mpan 651 . . 3  |-  ( C  e.  U_ x  e.  A  ( { x }  X.  B )  ->  E. x E. y  C  =  <. x ,  y
>. )
76pm4.71ri 614 . 2  |-  ( C  e.  U_ x  e.  A  ( { x }  X.  B )  <->  ( E. x E. y  C  = 
<. x ,  y >.  /\  C  e.  U_ x  e.  A  ( {
x }  X.  B
) ) )
8 nfiu1 3933 . . . 4  |-  F/_ x U_ x  e.  A  ( { x }  X.  B )
98nfel2 2431 . . 3  |-  F/ x  C  e.  U_ x  e.  A  ( { x }  X.  B )
10919.41 1815 . 2  |-  ( E. x ( E. y  C  =  <. x ,  y >.  /\  C  e. 
U_ x  e.  A  ( { x }  X.  B ) )  <->  ( E. x E. y  C  = 
<. x ,  y >.  /\  C  e.  U_ x  e.  A  ( {
x }  X.  B
) ) )
11 19.41v 1842 . . . 4  |-  ( E. y ( C  = 
<. x ,  y >.  /\  C  e.  U_ x  e.  A  ( {
x }  X.  B
) )  <->  ( E. y  C  =  <. x ,  y >.  /\  C  e.  U_ x  e.  A  ( { x }  X.  B ) ) )
12 eleq1 2343 . . . . . . 7  |-  ( C  =  <. x ,  y
>.  ->  ( C  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  <. x ,  y >.  e.  U_ x  e.  A  ( {
x }  X.  B
) ) )
13 opeliunxp 4740 . . . . . . 7  |-  ( <.
x ,  y >.  e.  U_ x  e.  A  ( { x }  X.  B )  <->  ( x  e.  A  /\  y  e.  B ) )
1412, 13syl6bb 252 . . . . . 6  |-  ( C  =  <. x ,  y
>.  ->  ( C  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  ( x  e.  A  /\  y  e.  B ) ) )
1514pm5.32i 618 . . . . 5  |-  ( ( C  =  <. x ,  y >.  /\  C  e.  U_ x  e.  A  ( { x }  X.  B ) )  <->  ( C  =  <. x ,  y
>.  /\  ( x  e.  A  /\  y  e.  B ) ) )
1615exbii 1569 . . . 4  |-  ( E. y ( C  = 
<. x ,  y >.  /\  C  e.  U_ x  e.  A  ( {
x }  X.  B
) )  <->  E. y
( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
) )
1711, 16bitr3i 242 . . 3  |-  ( ( E. y  C  = 
<. x ,  y >.  /\  C  e.  U_ x  e.  A  ( {
x }  X.  B
) )  <->  E. y
( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
) )
1817exbii 1569 . 2  |-  ( E. x ( E. y  C  =  <. x ,  y >.  /\  C  e. 
U_ x  e.  A  ( { x }  X.  B ) )  <->  E. x E. y ( C  = 
<. x ,  y >.  /\  ( x  e.  A  /\  y  e.  B
) ) )
197, 10, 183bitr2i 264 1  |-  ( C  e.  U_ x  e.  A  ( { x }  X.  B )  <->  E. x E. y ( C  = 
<. x ,  y >.  /\  ( x  e.  A  /\  y  e.  B
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543   {csn 3640   <.cop 3643   U_ciun 3905    X. cxp 4687   Rel wrel 4694
This theorem is referenced by:  raliunxp  4825  dfmpt3  5366  mpt2mptx  5938  fsumcom2  12237  isfunc  13738  gsum2d2  15225  dprd2d2  15279  fsumvma  20452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-iun 3907  df-opab 4078  df-xp 4695  df-rel 4696
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