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Theorem eliunxp 5012
Description: Membership in a union of cross products. Analogue of elxp 4895 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 4983 . . . . . 6  |-  Rel  ( { x }  X.  B )
21rgenw 2773 . . . . 5  |-  A. x  e.  A  Rel  ( { x }  X.  B
)
3 reliun 4995 . . . . 5  |-  ( Rel  U_ x  e.  A  ( { x }  X.  B )  <->  A. x  e.  A  Rel  ( { x }  X.  B
) )
42, 3mpbir 201 . . . 4  |-  Rel  U_ x  e.  A  ( {
x }  X.  B
)
5 elrel 4978 . . . 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 652 . . 3  |-  ( C  e.  U_ x  e.  A  ( { x }  X.  B )  ->  E. x E. y  C  =  <. x ,  y
>. )
76pm4.71ri 615 . 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 4121 . . . 4  |-  F/_ x U_ x  e.  A  ( { x }  X.  B )
98nfel2 2584 . . 3  |-  F/ x  C  e.  U_ x  e.  A  ( { x }  X.  B )
10919.41 1900 . 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 1924 . . . 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 2496 . . . . . . 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 4929 . . . . . . 7  |-  ( <.
x ,  y >.  e.  U_ x  e.  A  ( { x }  X.  B )  <->  ( x  e.  A  /\  y  e.  B ) )
1412, 13syl6bb 253 . . . . . 6  |-  ( C  =  <. x ,  y
>.  ->  ( C  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  ( x  e.  A  /\  y  e.  B ) ) )
1514pm5.32i 619 . . . . 5  |-  ( ( C  =  <. x ,  y >.  /\  C  e.  U_ x  e.  A  ( { x }  X.  B ) )  <->  ( C  =  <. x ,  y
>.  /\  ( x  e.  A  /\  y  e.  B ) ) )
1615exbii 1592 . . . 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 243 . . 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 1592 . 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 265 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 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   A.wral 2705   {csn 3814   <.cop 3817   U_ciun 4093    X. cxp 4876   Rel wrel 4883
This theorem is referenced by:  raliunxp  5014  dfmpt3  5567  mpt2mptx  6164  fsumcom2  12558  isfunc  14061  gsum2d2  15548  dprd2d2  15602  fsumvma  20997  fprodcom2  25308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-iun 4095  df-opab 4267  df-xp 4884  df-rel 4885
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