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Theorem iunxpf 4427
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/_ C
iunxpf.2  F/_ C
iunxpf.3  F/_ D
iunxpf.4  <. , 
>.  C  D
Assertion
Ref Expression
iunxpf  U_  X.  C  U_  U_  D
Distinct variable groups:   ,,   ,,,
Allowed substitution hints:   ()    C(,,)    D(,,)

Proof of Theorem iunxpf
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 iunxpf.1 . . . . 5  F/_ C
21nfcri 2169 . . . 4  F/  C
3 iunxpf.2 . . . . 5  F/_ C
43nfcri 2169 . . . 4  F/  C
5 iunxpf.3 . . . . 5  F/_ D
65nfcri 2169 . . . 4  F/  D
7 iunxpf.4 . . . . 5  <. , 
>.  C  D
87eleq2d 2104 . . . 4  <. , 
>.  C  D
92, 4, 6, 8rexxpf 4426 . . 3  X.  C  D
10 eliun 3652 . . 3  U_  X.  C  X.  C
11 eliun 3652 . . . 4  U_  U_  D  U_  D
12 eliun 3652 . . . . 5  U_  D  D
1312rexbii 2325 . . . 4  U_  D  D
1411, 13bitri 173 . . 3  U_  U_  D  D
159, 10, 143bitr4i 201 . 2  U_  X.  C  U_  U_  D
1615eqriv 2034 1  U_  X.  C  U_  U_  D
Colors of variables: wff set class
Syntax hints:   wi 4   wceq 1242   wcel 1390   F/_wnfc 2162  wrex 2301   <.cop 3370   U_ciun 3648    X. cxp 4286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iun 3650  df-opab 3810  df-xp 4294  df-rel 4295
This theorem is referenced by:  dfmpt2  5786
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