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.

Step	Hyp	Ref	Expression
1		iunxpf.1	. . . . 5
2	1	nfcri 2612	. . . 4
3		iunxpf.2	. . . . 5
4	3	nfcri 2612	. . . 4
5		iunxpf.3	. . . . 5
6	5	nfcri 2612	. . . 4
7		iunxpf.4	. . . . 5
8	7	eleq2d 2527	. . . 4
9	2, 4, 6, 8	rexxpf 5155	. . 3
10		eliun 4335	. . 3
11		eliun 4335	. . . 4
12		eliun 4335	. . . . 5
13	12	rexbii 2959	. . . 4
14	11, 13	bitri 249	. . 3
15	9, 10, 14	3bitr4i 277	. 2
16	15	eqriv 2453	1

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