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Theorem opeliunxp 4929
Description: Membership in a union of cross products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)
Assertion
Ref Expression
opeliunxp  |-  ( <.
x ,  C >.  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  ( x  e.  A  /\  C  e.  B ) )

Proof of Theorem opeliunxp
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-iun 4095 . . 3  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  { y  |  E. x  e.  A  y  e.  ( { x }  X.  B ) }
21eleq2i 2500 . 2  |-  ( <.
x ,  C >.  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  <. x ,  C >.  e.  { y  |  E. x  e.  A  y  e.  ( { x }  X.  B ) } )
3 opex 4427 . . 3  |-  <. x ,  C >.  e.  _V
4 df-rex 2711 . . . . 5  |-  ( E. x  e.  A  y  e.  ( { x }  X.  B )  <->  E. x
( x  e.  A  /\  y  e.  ( { x }  X.  B ) ) )
5 nfv 1629 . . . . . 6  |-  F/ z ( x  e.  A  /\  y  e.  ( { x }  X.  B ) )
6 nfs1v 2182 . . . . . . 7  |-  F/ x [ z  /  x ] x  e.  A
7 nfcv 2572 . . . . . . . . 9  |-  F/_ x { z }
8 nfcsb1v 3283 . . . . . . . . 9  |-  F/_ x [_ z  /  x ]_ B
97, 8nfxp 4904 . . . . . . . 8  |-  F/_ x
( { z }  X.  [_ z  /  x ]_ B )
109nfcri 2566 . . . . . . 7  |-  F/ x  y  e.  ( {
z }  X.  [_ z  /  x ]_ B
)
116, 10nfan 1846 . . . . . 6  |-  F/ x
( [ z  /  x ] x  e.  A  /\  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) )
12 sbequ12 1944 . . . . . . 7  |-  ( x  =  z  ->  (
x  e.  A  <->  [ z  /  x ] x  e.  A ) )
13 sneq 3825 . . . . . . . . 9  |-  ( x  =  z  ->  { x }  =  { z } )
14 csbeq1a 3259 . . . . . . . . 9  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
1513, 14xpeq12d 4903 . . . . . . . 8  |-  ( x  =  z  ->  ( { x }  X.  B )  =  ( { z }  X.  [_ z  /  x ]_ B ) )
1615eleq2d 2503 . . . . . . 7  |-  ( x  =  z  ->  (
y  e.  ( { x }  X.  B
)  <->  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
1712, 16anbi12d 692 . . . . . 6  |-  ( x  =  z  ->  (
( x  e.  A  /\  y  e.  ( { x }  X.  B ) )  <->  ( [
z  /  x ]
x  e.  A  /\  y  e.  ( {
z }  X.  [_ z  /  x ]_ B
) ) ) )
185, 11, 17cbvex 1983 . . . . 5  |-  ( E. x ( x  e.  A  /\  y  e.  ( { x }  X.  B ) )  <->  E. z
( [ z  /  x ] x  e.  A  /\  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
194, 18bitri 241 . . . 4  |-  ( E. x  e.  A  y  e.  ( { x }  X.  B )  <->  E. z
( [ z  /  x ] x  e.  A  /\  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
20 eleq1 2496 . . . . . 6  |-  ( y  =  <. x ,  C >.  ->  ( y  e.  ( { z }  X.  [_ z  /  x ]_ B )  <->  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
2120anbi2d 685 . . . . 5  |-  ( y  =  <. x ,  C >.  ->  ( ( [ z  /  x ]
x  e.  A  /\  y  e.  ( {
z }  X.  [_ z  /  x ]_ B
) )  <->  ( [
z  /  x ]
x  e.  A  /\  <.
x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) ) )
2221exbidv 1636 . . . 4  |-  ( y  =  <. x ,  C >.  ->  ( E. z
( [ z  /  x ] x  e.  A  /\  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) )  <->  E. z
( [ z  /  x ] x  e.  A  /\  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) ) )
2319, 22syl5bb 249 . . 3  |-  ( y  =  <. x ,  C >.  ->  ( E. x  e.  A  y  e.  ( { x }  X.  B )  <->  E. z
( [ z  /  x ] x  e.  A  /\  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) ) )
243, 23elab 3082 . 2  |-  ( <.
x ,  C >.  e. 
{ y  |  E. x  e.  A  y  e.  ( { x }  X.  B ) }  <->  E. z
( [ z  /  x ] x  e.  A  /\  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
25 opelxp 4908 . . . . . 6  |-  ( <.
x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B )  <->  ( x  e.  { z }  /\  C  e.  [_ z  /  x ]_ B ) )
2625anbi2i 676 . . . . 5  |-  ( ( [ z  /  x ] x  e.  A  /\  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) )  <->  ( [ z  /  x ] x  e.  A  /\  (
x  e.  { z }  /\  C  e. 
[_ z  /  x ]_ B ) ) )
27 an12 773 . . . . 5  |-  ( ( [ z  /  x ] x  e.  A  /\  ( x  e.  {
z }  /\  C  e.  [_ z  /  x ]_ B ) )  <->  ( x  e.  { z }  /\  ( [ z  /  x ] x  e.  A  /\  C  e.  [_ z  /  x ]_ B ) ) )
28 elsn 3829 . . . . . . 7  |-  ( x  e.  { z }  <-> 
x  =  z )
29 equcom 1692 . . . . . . 7  |-  ( x  =  z  <->  z  =  x )
3028, 29bitri 241 . . . . . 6  |-  ( x  e.  { z }  <-> 
z  =  x )
3130anbi1i 677 . . . . 5  |-  ( ( x  e.  { z }  /\  ( [ z  /  x ]
x  e.  A  /\  C  e.  [_ z  /  x ]_ B ) )  <-> 
( z  =  x  /\  ( [ z  /  x ] x  e.  A  /\  C  e. 
[_ z  /  x ]_ B ) ) )
3226, 27, 313bitri 263 . . . 4  |-  ( ( [ z  /  x ] x  e.  A  /\  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) )  <->  ( z  =  x  /\  ( [ z  /  x ]
x  e.  A  /\  C  e.  [_ z  /  x ]_ B ) ) )
3332exbii 1592 . . 3  |-  ( E. z ( [ z  /  x ] x  e.  A  /\  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) )  <->  E. z
( z  =  x  /\  ( [ z  /  x ] x  e.  A  /\  C  e. 
[_ z  /  x ]_ B ) ) )
34 vex 2959 . . . 4  |-  x  e. 
_V
35 sbequ12r 1945 . . . . 5  |-  ( z  =  x  ->  ( [ z  /  x ] x  e.  A  <->  x  e.  A ) )
3614equcoms 1693 . . . . . . 7  |-  ( z  =  x  ->  B  =  [_ z  /  x ]_ B )
3736eqcomd 2441 . . . . . 6  |-  ( z  =  x  ->  [_ z  /  x ]_ B  =  B )
3837eleq2d 2503 . . . . 5  |-  ( z  =  x  ->  ( C  e.  [_ z  /  x ]_ B  <->  C  e.  B ) )
3935, 38anbi12d 692 . . . 4  |-  ( z  =  x  ->  (
( [ z  /  x ] x  e.  A  /\  C  e.  [_ z  /  x ]_ B )  <-> 
( x  e.  A  /\  C  e.  B
) ) )
4034, 39ceqsexv 2991 . . 3  |-  ( E. z ( z  =  x  /\  ( [ z  /  x ]
x  e.  A  /\  C  e.  [_ z  /  x ]_ B ) )  <-> 
( x  e.  A  /\  C  e.  B
) )
4133, 40bitri 241 . 2  |-  ( E. z ( [ z  /  x ] x  e.  A  /\  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) )  <->  ( x  e.  A  /\  C  e.  B ) )
422, 24, 413bitri 263 1  |-  ( <.
x ,  C >.  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  ( x  e.  A  /\  C  e.  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652   [wsb 1658    e. wcel 1725   {cab 2422   E.wrex 2706   [_csb 3251   {csn 3814   <.cop 3817   U_ciun 4093    X. cxp 4876
This theorem is referenced by:  eliunxp  5012  opeliunxp2  5013  gsum2d2lem  15547  gsum2d2  15548  gsumcom2  15549  dprdval  15561  ptbasfi  17613  cnextfun  18095  cnextfvval  18096  cnextf  18097  dvbsss  19789
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
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