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Theorem opeliunxp 4693
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 2760 . 2  |-  ( <.
x ,  C >.  e. 
U_ x  e.  A  ( { x }  X.  B )  ->  <. x ,  C >.  e.  _V )
2 opexg 4240 . 2  |-  ( ( x  e.  A  /\  C  e.  B )  -> 
<. x ,  C >.  e. 
_V )
3 df-rex 2471 . . . . . 6  |-  ( E. x  e.  A  y  e.  ( { x }  X.  B )  <->  E. x
( x  e.  A  /\  y  e.  ( { x }  X.  B ) ) )
4 nfv 1538 . . . . . . 7  |-  F/ z ( x  e.  A  /\  y  e.  ( { x }  X.  B ) )
5 nfs1v 1949 . . . . . . . 8  |-  F/ x [ z  /  x ] x  e.  A
6 nfcv 2329 . . . . . . . . . 10  |-  F/_ x { z }
7 nfcsb1v 3102 . . . . . . . . . 10  |-  F/_ x [_ z  /  x ]_ B
86, 7nfxp 4665 . . . . . . . . 9  |-  F/_ x
( { z }  X.  [_ z  /  x ]_ B )
98nfcri 2323 . . . . . . . 8  |-  F/ x  y  e.  ( {
z }  X.  [_ z  /  x ]_ B
)
105, 9nfan 1575 . . . . . . 7  |-  F/ x
( [ z  /  x ] x  e.  A  /\  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) )
11 sbequ12 1781 . . . . . . . 8  |-  ( x  =  z  ->  (
x  e.  A  <->  [ z  /  x ] x  e.  A ) )
12 sneq 3615 . . . . . . . . . 10  |-  ( x  =  z  ->  { x }  =  { z } )
13 csbeq1a 3078 . . . . . . . . . 10  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
1412, 13xpeq12d 4663 . . . . . . . . 9  |-  ( x  =  z  ->  ( { x }  X.  B )  =  ( { z }  X.  [_ z  /  x ]_ B ) )
1514eleq2d 2257 . . . . . . . 8  |-  ( x  =  z  ->  (
y  e.  ( { x }  X.  B
)  <->  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
1611, 15anbi12d 473 . . . . . . 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 1766 . . . . . 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 184 . . . . 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 2250 . . . . . . 7  |-  ( y  =  <. x ,  C >.  ->  ( y  e.  ( { z }  X.  [_ z  /  x ]_ B )  <->  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
2019anbi2d 464 . . . . . 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 1835 . . . . 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, 21bitrid 192 . . . 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 3900 . . . 4  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  { y  |  E. x  e.  A  y  e.  ( { x }  X.  B ) }
2422, 23elab2g 2896 . . 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 4668 . . . . . . 7  |-  ( <.
x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B )  <->  ( x  e.  { z }  /\  C  e.  [_ z  /  x ]_ B ) )
2625anbi2i 457 . . . . . 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 561 . . . . . 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 3621 . . . . . . . 8  |-  ( x  e.  { z }  <-> 
x  =  z )
29 equcom 1716 . . . . . . . 8  |-  ( x  =  z  <->  z  =  x )
3028, 29bitri 184 . . . . . . 7  |-  ( x  e.  { z }  <-> 
z  =  x )
3130anbi1i 458 . . . . . 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 206 . . . . 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 1615 . . . 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 2752 . . . . 5  |-  x  e. 
_V
35 sbequ12r 1782 . . . . . 6  |-  ( z  =  x  ->  ( [ z  /  x ] x  e.  A  <->  x  e.  A ) )
3613equcoms 1718 . . . . . . . 8  |-  ( z  =  x  ->  B  =  [_ z  /  x ]_ B )
3736eqcomd 2193 . . . . . . 7  |-  ( z  =  x  ->  [_ z  /  x ]_ B  =  B )
3837eleq2d 2257 . . . . . 6  |-  ( z  =  x  ->  ( C  e.  [_ z  /  x ]_ B  <->  C  e.  B ) )
3935, 38anbi12d 473 . . . . 5  |-  ( z  =  x  ->  (
( [ z  /  x ] x  e.  A  /\  C  e.  [_ z  /  x ]_ B )  <-> 
( x  e.  A  /\  C  e.  B
) ) )
4034, 39ceqsexv 2788 . . . 4  |-  ( E. z ( z  =  x  /\  ( [ z  /  x ]
x  e.  A  /\  C  e.  [_ z  /  x ]_ B ) )  <-> 
( x  e.  A  /\  C  e.  B
) )
4133, 40bitri 184 . . 3  |-  ( E. z ( [ z  /  x ] x  e.  A  /\  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) )  <->  ( x  e.  A  /\  C  e.  B ) )
4224, 41bitrdi 196 . 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 705 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 104    <-> wb 105    = wceq 1363   E.wex 1502   [wsb 1772    e. wcel 2158   E.wrex 2466   _Vcvv 2749   [_csb 3069   {csn 3604   <.cop 3607   U_ciun 3898    X. cxp 4636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-iun 3900  df-opab 4077  df-xp 4644
This theorem is referenced by:  eliunxp  4778  opeliunxp2  4779  opeliunxp2f  6252
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