Theorem coiun 5522

Description: Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)

Assertion

Ref	Expression
coiun

Distinct variable group:   ,

Proof of Theorem coiun

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relco 5510	. 2
2		reliun 5128	. . 3
3		relco 5510	. . . 4
4	3	a1i 11	. . 3
5	2, 4	mprgbir 2821	. 2
6		eliun 4335	. . . . . . . 8
7		df-br 4453	. . . . . . . 8
8		df-br 4453	. . . . . . . . 9
9	8	rexbii 2959	. . . . . . . 8
10	6, 7, 9	3bitr4i 277	. . . . . . 7
11	10	anbi1i 695	. . . . . 6
12		r19.41v 3009	. . . . . 6
13	11, 12	bitr4i 252	. . . . 5
14	13	exbii 1667	. . . 4
15		rexcom4 3129	. . . 4
16	14, 15	bitr4i 252	. . 3
17		vex 3112	. . . 4
18		vex 3112	. . . 4
19	17, 18	opelco 5179	. . 3
20		eliun 4335	. . . 4
21	17, 18	opelco 5179	. . . . 5
22	21	rexbii 2959	. . . 4
23	20, 22	bitri 249	. . 3
24	16, 19, 23	3bitr4i 277	. 2
25	1, 5, 24	eqrelriiv 5102	1

Syntax hints:  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  E.wrex 2808  <.cop 4035  U_ciun 4330   class class class wbr 4452  o.ccom 5008  Relwrel 5009

This theorem is referenced by:  fparlem3  6902  fparlem4  6903

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-eu 2286  df-mo 2287  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-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-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-co 5013