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Theorem iunxpf 4869
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 2446 . . . 4  |-  F/ y  w  e.  C
3 iunxpf.2 . . . . 5  |-  F/_ z C
43nfcri 2446 . . . 4  |-  F/ z  w  e.  C
5 iunxpf.3 . . . . 5  |-  F/_ x D
65nfcri 2446 . . . 4  |-  F/ x  w  e.  D
7 iunxpf.4 . . . . 5  |-  ( x  =  <. y ,  z
>.  ->  C  =  D )
87eleq2d 2383 . . . 4  |-  ( x  =  <. y ,  z
>.  ->  ( w  e.  C  <->  w  e.  D
) )
92, 4, 6, 8rexxpf 4868 . . 3  |-  ( E. x  e.  ( A  X.  B ) w  e.  C  <->  E. y  e.  A  E. z  e.  B  w  e.  D )
10 eliun 3946 . . 3  |-  ( w  e.  U_ x  e.  ( A  X.  B
) C  <->  E. x  e.  ( A  X.  B
) w  e.  C
)
11 eliun 3946 . . . 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 3946 . . . . 5  |-  ( w  e.  U_ z  e.  B  D  <->  E. z  e.  B  w  e.  D )
1312rexbii 2602 . . . 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 240 . . 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 268 . 2  |-  ( w  e.  U_ x  e.  ( A  X.  B
) C  <->  w  e.  U_ y  e.  A  U_ z  e.  B  D
)
1615eqriv 2313 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 1633    e. wcel 1701   F/_wnfc 2439   E.wrex 2578   <.cop 3677   U_ciun 3942    X. cxp 4724
This theorem is referenced by:  dfmpt2  6251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-iun 3944  df-opab 4115  df-xp 4732  df-rel 4733
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