Theorem coiun1 37944

Description: Composition with an indexed union. Proof analgous to that of coiun 5645. (Contributed by RP, 20-Jun-2020.)

Assertion

Ref	Expression
coiun1	|- ( U_ x e. C A o. B ) = U_ x e. C ( A o. B )

Distinct variable group:   x, B

Allowed substitution hints:   A( x)   C( x)

Proof of Theorem coiun1

Dummy variables y w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relco 5633	. 2 |- Rel ( U_ x e. C A o. B )
2		reliun 5239	. . 3 |- ( Rel U_ x e. C ( A o. B ) <-> A. x e. C Rel ( A o. B ) )
3		relco 5633	. . . 4 |- Rel ( A o. B )
4	3	a1i 11	. . 3 |- ( x e. C -> Rel ( A o. B ) )
5	2, 4	mprgbir 2927	. 2 |- Rel U_ x e. C ( A o. B )
6		eliun 4524	. . . . . . . 8 |- ( <. w , z >. e. U_ x e. C A <-> E. x e. C <. w , z >. e. A )
7		df-br 4654	. . . . . . . 8 |- ( w U_ x e. C A z <-> <. w , z >. e. U_ x e. C A )
8		df-br 4654	. . . . . . . . 9 |- ( w A z <-> <. w , z >. e. A )
9	8	rexbii 3041	. . . . . . . 8 |- ( E. x e. C w A z <-> E. x e. C <. w , z >. e. A )
10	6, 7, 9	3bitr4i 292	. . . . . . 7 |- ( w U_ x e. C A z <-> E. x e. C w A z )
11	10	anbi2i 730	. . . . . 6 |- ( ( y B w /\ w U_ x e. C A z ) <-> ( y B w /\ E. x e. C w A z ) )
12		r19.42v 3092	. . . . . 6 |- ( E. x e. C ( y B w /\ w A z ) <-> ( y B w /\ E. x e. C w A z ) )
13	11, 12	bitr4i 267	. . . . 5 |- ( ( y B w /\ w U_ x e. C A z ) <-> E. x e. C ( y B w /\ w A z ) )
14	13	exbii 1774	. . . 4 |- ( E. w ( y B w /\ w U_ x e. C A z ) <-> E. w E. x e. C ( y B w /\ w A z ) )
15		rexcom4 3225	. . . 4 |- ( E. x e. C E. w ( y B w /\ w A z ) <-> E. w E. x e. C ( y B w /\ w A z ) )
16	14, 15	bitr4i 267	. . 3 |- ( E. w ( y B w /\ w U_ x e. C A z ) <-> E. x e. C E. w ( y B w /\ w A z ) )
17		vex 3203	. . . 4 |- y e. _V
18		vex 3203	. . . 4 |- z e. _V
19	17, 18	opelco 5293	. . 3 |- ( <. y , z >. e. ( U_ x e. C A o. B ) <-> E. w ( y B w /\ w U_ x e. C A z ) )
20		eliun 4524	. . . 4 |- ( <. y , z >. e. U_ x e. C ( A o. B ) <-> E. x e. C <. y , z >. e. ( A o. B ) )
21	17, 18	opelco 5293	. . . . 5 |- ( <. y , z >. e. ( A o. B ) <-> E. w ( y B w /\ w A z ) )
22	21	rexbii 3041	. . . 4 |- ( E. x e. C <. y , z >. e. ( A o. B ) <-> E. x e. C E. w ( y B w /\ w A z ) )
23	20, 22	bitri 264	. . 3 |- ( <. y , z >. e. U_ x e. C ( A o. B ) <-> E. x e. C E. w ( y B w /\ w A z ) )
24	16, 19, 23	3bitr4i 292	. 2 |- ( <. y , z >. e. ( U_ x e. C A o. B ) <-> <. y , z >. e. U_ x e. C ( A o. B ) )
25	1, 5, 24	eqrelriiv 5214	1 |- ( U_ x e. C A o. B ) = U_ x e. C ( A o. B )

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   <.cop 4183   U_ciun 4520   class class class wbr 4653   o. ccom 5118   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-iun 4522  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-co 5123

This theorem is referenced by:  trclfvcom  38015  trclfvdecomr  38020  cotrclrcl  38034