Theorem iunpw 6614

Description: An indexed union of a power class in terms of the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)

Hypothesis

Ref	Expression
iunpw.1

Assertion

Ref	Expression
iunpw

Distinct variable group:   ,

Proof of Theorem iunpw

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq2 3525	. . . . . . . 8
2	1	biimprcd 225	. . . . . . 7
3	2	reximdv 2931	. . . . . 6
4	3	com12 31	. . . . 5
5		ssiun 4372	. . . . . 6
6		uniiun 4383	. . . . . 6
7	5, 6	syl6sseqr 3550	. . . . 5
8	4, 7	impbid1 203	. . . 4
9		selpw 4019	. . . 4
10		eliun 4335	. . . . 5
11		selpw 4019	. . . . . 6
12	11	rexbii 2959	. . . . 5
13	10, 12	bitri 249	. . . 4
14	8, 9, 13	3bitr4g 288	. . 3
15	14	eqrdv 2454	. 2
16		ssid 3522	. . . . 5
17		iunpw.1	. . . . . . . 8
18	17	uniex 6596	. . . . . . 7
19	18	elpw 4018	. . . . . 6
20		eleq2 2530	. . . . . 6
21	19, 20	syl5bbr 259	. . . . 5
22	16, 21	mpbii 211	. . . 4
23		eliun 4335	. . . 4
24	22, 23	sylib 196	. . 3
25		elssuni 4279	. . . . . . 7
26		elpwi 4021	. . . . . . 7
27	25, 26	anim12i 566	. . . . . 6
28		eqss 3518	. . . . . 6
29	27, 28	sylibr 212	. . . . 5
30	29	ex 434	. . . 4
31	30	reximia 2923	. . 3
32	24, 31	syl 16	. 2
33	15, 32	impbii 188	1

Syntax hints:  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  E.wrex 2808  cvv 3109  C_wss 3475  ~Pcpw 4012  U.cuni 4249  U_ciun 4330

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-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-un 6592

This theorem depends on definitions:  df-bi 185  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-ss 3489  df-pw 4014  df-uni 4250  df-iun 4332