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Theorem opeliunxp 4564
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 elex 2671 . 2  |-  ( <.
x ,  C >.  e. 
U_ x  e.  A  ( { x }  X.  B )  ->  <. x ,  C >.  e.  _V )
2 opexg 4120 . 2  |-  ( ( x  e.  A  /\  C  e.  B )  -> 
<. x ,  C >.  e. 
_V )
3 df-rex 2399 . . . . . 6  |-  ( E. x  e.  A  y  e.  ( { x }  X.  B )  <->  E. x
( x  e.  A  /\  y  e.  ( { x }  X.  B ) ) )
4 nfv 1493 . . . . . . 7  |-  F/ z ( x  e.  A  /\  y  e.  ( { x }  X.  B ) )
5 nfs1v 1892 . . . . . . . 8  |-  F/ x [ z  /  x ] x  e.  A
6 nfcv 2258 . . . . . . . . . 10  |-  F/_ x { z }
7 nfcsb1v 3005 . . . . . . . . . 10  |-  F/_ x [_ z  /  x ]_ B
86, 7nfxp 4536 . . . . . . . . 9  |-  F/_ x
( { z }  X.  [_ z  /  x ]_ B )
98nfcri 2252 . . . . . . . 8  |-  F/ x  y  e.  ( {
z }  X.  [_ z  /  x ]_ B
)
105, 9nfan 1529 . . . . . . 7  |-  F/ x
( [ z  /  x ] x  e.  A  /\  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) )
11 sbequ12 1729 . . . . . . . 8  |-  ( x  =  z  ->  (
x  e.  A  <->  [ z  /  x ] x  e.  A ) )
12 sneq 3508 . . . . . . . . . 10  |-  ( x  =  z  ->  { x }  =  { z } )
13 csbeq1a 2983 . . . . . . . . . 10  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
1412, 13xpeq12d 4534 . . . . . . . . 9  |-  ( x  =  z  ->  ( { x }  X.  B )  =  ( { z }  X.  [_ z  /  x ]_ B ) )
1514eleq2d 2187 . . . . . . . 8  |-  ( x  =  z  ->  (
y  e.  ( { x }  X.  B
)  <->  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
1611, 15anbi12d 464 . . . . . . 7  |-  ( x  =  z  ->  (
( x  e.  A  /\  y  e.  ( { x }  X.  B ) )  <->  ( [
z  /  x ]
x  e.  A  /\  y  e.  ( {
z }  X.  [_ z  /  x ]_ B
) ) ) )
174, 10, 16cbvex 1714 . . . . . 6  |-  ( E. x ( x  e.  A  /\  y  e.  ( { x }  X.  B ) )  <->  E. z
( [ z  /  x ] x  e.  A  /\  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
183, 17bitri 183 . . . . 5  |-  ( E. x  e.  A  y  e.  ( { x }  X.  B )  <->  E. z
( [ z  /  x ] x  e.  A  /\  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
19 eleq1 2180 . . . . . . 7  |-  ( y  =  <. x ,  C >.  ->  ( y  e.  ( { z }  X.  [_ z  /  x ]_ B )  <->  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
2019anbi2d 459 . . . . . 6  |-  ( 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 ) ) ) )
2120exbidv 1781 . . . . 5  |-  ( 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 ) ) ) )
2218, 21syl5bb 191 . . . 4  |-  ( 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 ) ) ) )
23 df-iun 3785 . . . 4  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  { y  |  E. x  e.  A  y  e.  ( { x }  X.  B ) }
2422, 23elab2g 2804 . . 3  |-  ( <.
x ,  C >.  e. 
_V  ->  ( <. x ,  C >.  e.  U_ x  e.  A  ( {
x }  X.  B
)  <->  E. z ( [ z  /  x ]
x  e.  A  /\  <.
x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) ) )
25 opelxp 4539 . . . . . . 7  |-  ( <.
x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B )  <->  ( x  e.  { z }  /\  C  e.  [_ z  /  x ]_ B ) )
2625anbi2i 452 . . . . . 6  |-  ( ( [ 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 535 . . . . . 6  |-  ( ( [ 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 velsn 3514 . . . . . . . 8  |-  ( x  e.  { z }  <-> 
x  =  z )
29 equcom 1667 . . . . . . . 8  |-  ( x  =  z  <->  z  =  x )
3028, 29bitri 183 . . . . . . 7  |-  ( x  e.  { z }  <-> 
z  =  x )
3130anbi1i 453 . . . . . 6  |-  ( ( 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 205 . . . . 5  |-  ( ( [ 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 1569 . . . 4  |-  ( 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 2663 . . . . 5  |-  x  e. 
_V
35 sbequ12r 1730 . . . . . 6  |-  ( z  =  x  ->  ( [ z  /  x ] x  e.  A  <->  x  e.  A ) )
3613equcoms 1669 . . . . . . . 8  |-  ( z  =  x  ->  B  =  [_ z  /  x ]_ B )
3736eqcomd 2123 . . . . . . 7  |-  ( z  =  x  ->  [_ z  /  x ]_ B  =  B )
3837eleq2d 2187 . . . . . 6  |-  ( z  =  x  ->  ( C  e.  [_ z  /  x ]_ B  <->  C  e.  B ) )
3935, 38anbi12d 464 . . . . 5  |-  ( z  =  x  ->  (
( [ z  /  x ] x  e.  A  /\  C  e.  [_ z  /  x ]_ B )  <-> 
( x  e.  A  /\  C  e.  B
) ) )
4034, 39ceqsexv 2699 . . . 4  |-  ( E. z ( z  =  x  /\  ( [ z  /  x ]
x  e.  A  /\  C  e.  [_ z  /  x ]_ B ) )  <-> 
( x  e.  A  /\  C  e.  B
) )
4133, 40bitri 183 . . 3  |-  ( E. z ( [ z  /  x ] x  e.  A  /\  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) )  <->  ( x  e.  A  /\  C  e.  B ) )
4224, 41syl6bb 195 . 2  |-  ( <.
x ,  C >.  e. 
_V  ->  ( <. x ,  C >.  e.  U_ x  e.  A  ( {
x }  X.  B
)  <->  ( x  e.  A  /\  C  e.  B ) ) )
431, 2, 42pm5.21nii 678 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:    /\ wa 103    <-> wb 104    = wceq 1316   E.wex 1453    e. wcel 1465   [wsb 1720   E.wrex 2394   _Vcvv 2660   [_csb 2975   {csn 3497   <.cop 3500   U_ciun 3783    X. cxp 4507
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-opab 3960  df-xp 4515
This theorem is referenced by:  eliunxp  4648  opeliunxp2  4649  opeliunxp2f  6103
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