Theorem inuni 4614

Description: The intersection of a union with a class is equal to the union of the intersections of each element of with . (Contributed by FL, 24-Mar-2007.)

Assertion

Ref	Expression
inuni

Distinct variable groups:   ,,   ,,

Proof of Theorem inuni

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni2 4253	. . . . 5
2	1	anbi1i 695	. . . 4
3		elin 3686	. . . 4
4		ancom 450	. . . . . . . 8
5		r19.41v 3009	. . . . . . . 8
6	4, 5	bitr4i 252	. . . . . . 7
7	6	exbii 1667	. . . . . 6
8		rexcom4 3129	. . . . . 6
9	7, 8	bitr4i 252	. . . . 5
10		vex 3112	. . . . . . . . . 10
11	10	inex1 4593	. . . . . . . . 9
12		eleq2 2530	. . . . . . . . 9
13	11, 12	ceqsexv 3146	. . . . . . . 8
14		elin 3686	. . . . . . . 8
15	13, 14	bitri 249	. . . . . . 7
16	15	rexbii 2959	. . . . . 6
17		r19.41v 3009	. . . . . 6
18	16, 17	bitri 249	. . . . 5
19	9, 18	bitri 249	. . . 4
20	2, 3, 19	3bitr4i 277	. . 3
21		eluniab 4260	. . 3
22	20, 21	bitr4i 252	. 2
23	22	eqriv 2453	1

Syntax hints:  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442  E.wrex 2808  i^icin 3474  U.cuni 4249

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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2812  df-rex 2813  df-v 3111  df-in 3482  df-uni 4250