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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
iunxpf.2
iunxpf.3
iunxpf.4
Assertion
Ref Expression
iunxpf
Distinct variable groups:   ,,   ,,,
Allowed substitution hints:   ()   (,,)   (,,)

Proof of Theorem iunxpf
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 iunxpf.1 . . . . 5
21nfcri 2568 . . . 4
3 iunxpf.2 . . . . 5
43nfcri 2568 . . . 4
5 iunxpf.3 . . . . 5
65nfcri 2568 . . . 4
7 iunxpf.4 . . . . 5
87eleq2d 2505 . . . 4
92, 4, 6, 8rexxpf 5022 . . 3
10 eliun 4099 . . 3
11 eliun 4099 . . . 4
12 eliun 4099 . . . . 5
1312rexbii 2732 . . . 4
1411, 13bitri 242 . . 3
159, 10, 143bitr4i 270 . 2
1615eqriv 2435 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1653   wcel 1726  wnfc 2561  wrex 2708  cop 3819  ciun 4095   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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