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Theorem iunxpf 5161
Description: Indexed union on a Cartesian 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 2612 . . . 4  |-  F/ y  w  e.  C
3 iunxpf.2 . . . . 5  |-  F/_ z C
43nfcri 2612 . . . 4  |-  F/ z  w  e.  C
5 iunxpf.3 . . . . 5  |-  F/_ x D
65nfcri 2612 . . . 4  |-  F/ x  w  e.  D
7 iunxpf.4 . . . . 5  |-  ( x  =  <. y ,  z
>.  ->  C  =  D )
87eleq2d 2527 . . . 4  |-  ( x  =  <. y ,  z
>.  ->  ( w  e.  C  <->  w  e.  D
) )
92, 4, 6, 8rexxpf 5160 . . 3  |-  ( E. x  e.  ( A  X.  B ) w  e.  C  <->  E. y  e.  A  E. z  e.  B  w  e.  D )
10 eliun 4337 . . 3  |-  ( w  e.  U_ x  e.  ( A  X.  B
) C  <->  E. x  e.  ( A  X.  B
) w  e.  C
)
11 eliun 4337 . . . 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 4337 . . . . 5  |-  ( w  e.  U_ z  e.  B  D  <->  E. z  e.  B  w  e.  D )
1312rexbii 2959 . . . 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 249 . . 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 277 . 2  |-  ( w  e.  U_ x  e.  ( A  X.  B
) C  <->  w  e.  U_ y  e.  A  U_ z  e.  B  D
)
1615eqriv 2453 1  |-  U_ x  e.  ( A  X.  B
) C  =  U_ y  e.  A  U_ z  e.  B  D
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   F/_wnfc 2605   E.wrex 2808   <.cop 4038   U_ciun 4332    X. cxp 5006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-iun 4334  df-opab 4516  df-xp 5014  df-rel 5015
This theorem is referenced by:  dfmpt2  6889
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