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Theorem iunxpf 5023
Description: Indexed union on a cross product is equals a double indexed union. The hypothesis specifies an implicit substitution. (Contributed by NM, 19-Dec-2008.)
Hypotheses
Ref Expression
iunxpf.1  |-  F/_ y C
iunxpf.2  |-  F/_ z C
iunxpf.3  |-  F/_ x D
iunxpf.4  |-  ( x  =  <. y ,  z
>.  ->  C  =  D )
Assertion
Ref Expression
iunxpf  |-  U_ x  e.  ( A  X.  B
) C  =  U_ y  e.  A  U_ z  e.  B  D
Distinct variable groups:    x, y, A    x, z, B, y
Allowed substitution hints:    A( z)    C( x, y, z)    D( x, y, z)

Proof of Theorem iunxpf
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 iunxpf.1 . . . . 5  |-  F/_ y C
21nfcri 2568 . . . 4  |-  F/ y  w  e.  C
3 iunxpf.2 . . . . 5  |-  F/_ z C
43nfcri 2568 . . . 4  |-  F/ z  w  e.  C
5 iunxpf.3 . . . . 5  |-  F/_ x D
65nfcri 2568 . . . 4  |-  F/ x  w  e.  D
7 iunxpf.4 . . . . 5  |-  ( x  =  <. y ,  z
>.  ->  C  =  D )
87eleq2d 2505 . . . 4  |-  ( x  =  <. y ,  z
>.  ->  ( w  e.  C  <->  w  e.  D
) )
92, 4, 6, 8rexxpf 5022 . . 3  |-  ( E. x  e.  ( A  X.  B ) w  e.  C  <->  E. y  e.  A  E. z  e.  B  w  e.  D )
10 eliun 4099 . . 3  |-  ( w  e.  U_ x  e.  ( A  X.  B
) C  <->  E. x  e.  ( A  X.  B
) w  e.  C
)
11 eliun 4099 . . . 4  |-  ( w  e.  U_ y  e.  A  U_ z  e.  B  D  <->  E. y  e.  A  w  e.  U_ z  e.  B  D
)
12 eliun 4099 . . . . 5  |-  ( w  e.  U_ z  e.  B  D  <->  E. z  e.  B  w  e.  D )
1312rexbii 2732 . . . 4  |-  ( E. y  e.  A  w  e.  U_ z  e.  B  D  <->  E. y  e.  A  E. z  e.  B  w  e.  D )
1411, 13bitri 242 . . 3  |-  ( w  e.  U_ y  e.  A  U_ z  e.  B  D  <->  E. y  e.  A  E. z  e.  B  w  e.  D )
159, 10, 143bitr4i 270 . 2  |-  ( w  e.  U_ x  e.  ( A  X.  B
) C  <->  w  e.  U_ y  e.  A  U_ z  e.  B  D
)
1615eqriv 2435 1  |-  U_ x  e.  ( A  X.  B
) C  =  U_ y  e.  A  U_ z  e.  B  D
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   F/_wnfc 2561   E.wrex 2708   <.cop 3819   U_ciun 4095    X. cxp 4878
This theorem is referenced by:  dfmpt2  6439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-iun 4097  df-opab 4269  df-xp 4886  df-rel 4887
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