Theorem coiun 5425

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

Assertion

Ref	Expression
coiun

Distinct variable group:   ,

Allowed substitution hints:   ()   ()

Proof of Theorem coiun

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

Step	Hyp	Ref	Expression
1		relco 5414	. 2
2		reliun 5037	. . 3
3		relco 5414	. . . 4
4	3	a1i 11	. . 3
5	2, 4	mprgbir 2783	. 2
6		eliun 4126	. . . . . . . 8
7		df-br 4244	. . . . . . . 8
8		df-br 4244	. . . . . . . . 9
9	8	rexbii 2737	. . . . . . . 8
10	6, 7, 9	3bitr4i 270	. . . . . . 7
11	10	anbi1i 678	. . . . . 6
12		r19.41v 2868	. . . . . 6
13	11, 12	bitr4i 245	. . . . 5
14	13	exbii 1593	. . . 4
15		rexcom4 2984	. . . 4
16	14, 15	bitr4i 245	. . 3
17		vex 2968	. . . 4
18		vex 2968	. . . 4
19	17, 18	opelco 5086	. . 3
20		eliun 4126	. . . 4
21	17, 18	opelco 5086	. . . . 5
22	21	rexbii 2737	. . . 4
23	20, 22	bitri 242	. . 3
24	16, 19, 23	3bitr4i 270	. 2
25	1, 5, 24	eqrelriiv 5012	1

Syntax hints:  /\wa 360  E.wex 1551  =wceq 1654  e.wcel 1728  E.wrex 2713  <.cop 3844  U_ciun 4122   class class class wbr 4243  o.ccom 4923  Relwrel 4924

This theorem is referenced by:  fparlem3  6498  fparlem4  6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4364  ax-nul 4372  ax-pr 4442

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 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3766  df-sn 3847  df-pr 3848  df-op 3850  df-iun 4124  df-br 4244  df-opab 4302  df-xp 4925  df-rel 4926  df-co 4928