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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
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 2612 . . . 4
3 iunxpf.2 . . . . 5
43nfcri 2612 . . . 4
5 iunxpf.3 . . . . 5
65nfcri 2612 . . . 4
7 iunxpf.4 . . . . 5
87eleq2d 2527 . . . 4
92, 4, 6, 8rexxpf 5160 . . 3
10 eliun 4337 . . 3
11 eliun 4337 . . . 4
12 eliun 4337 . . . . 5
1312rexbii 2959 . . . 4
1411, 13bitri 249 . . 3
159, 10, 143bitr4i 277 . 2
1615eqriv 2453 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1395   wcel 1819  wnfc 2605  wrex 2808  cop 4038  ciun 4332   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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