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Theorem iunxpf 5156
 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:   ,,   ,,,

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 5155 . . 3
10 eliun 4335 . . 3
11 eliun 4335 . . . 4
12 eliun 4335 . . . . 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  e.wcel 1818  F/_wnfc 2605  E.wrex 2808  <.cop 4035  U_ciun 4330  X.cxp 5002 This theorem is referenced by:  dfmpt2  6890 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  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 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-iun 4332  df-opab 4511  df-xp 5010  df-rel 5011
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