Theorem rnuni 3459

Description: The range of a union. Part of Exercise 8 of [Enderton] p. 41.

Assertion

Ref	Expression
rnuni	|- ran U. A = U_x e. A ran x

Distinct variable group:   x,A

Proof of Theorem rnuni

Step	Hyp	Ref	Expression
1		eluni 2506	. . . . . 6 |- (<.y, z>. e. U.A <-> E.x(<.y, z>. e. x /\ x e. A))
2	1	exbii 1051	. . . . 5 |- (E.y<.y, z>. e. U.A <-> E.yE.x(<.y, z>. e. x /\ x e. A))
3		excom 1046	. . . . 5 |- (E.yE.x(<.y, z>. e. x /\ x e. A) <-> E.xE.y(<.y, z>. e. x /\ x e. A))
4		ancom 435	. . . . . . 7 |- ((E.y<.y, z>. e. x /\ x e. A) <-> (x e. A /\ E.y<.y, z>. e. x))
5		19.41v 1305	. . . . . . 7 |- (E.y(<.y, z>. e. x /\ x e. A) <-> (E.y<.y, z>. e. x /\ x e. A))
6		visset 1813	. . . . . . . . 9 |- z e. V
7	6	elrn2 3349	. . . . . . . 8 |- (z e. ran x <-> E.y<.y, z>. e. x)
8	7	anbi2i 480	. . . . . . 7 |- ((x e. A /\ z e. ran x) <-> (x e. A /\ E.y<.y, z>. e. x))
9	4, 5, 8	3bitr4 183	. . . . . 6 |- (E.y(<.y, z>. e. x /\ x e. A) <-> (x e. A /\ z e. ran x))
10	9	exbii 1051	. . . . 5 |- (E.xE.y(<.y, z>. e. x /\ x e. A) <-> E.x(x e. A /\ z e. ran x))
11	2, 3, 10	3bitr 177	. . . 4 |- (E.y<.y, z>. e. U.A <-> E.x(x e. A /\ z e. ran x))
12		df-rex 1650	. . . 4 |- (E.x e. A z e. ran x <-> E.x(x e. A /\ z e. ran x))
13	11, 12	bitr4 176	. . 3 |- (E.y<.y, z>. e. U.A <-> E.x e. A z e. ran x)
14	6	elrn2 3349	. . 3 |- (z e. ran U. A <-> E.y<.y, z>. e. U.A)
15		eliun 2570	. . 3 |- (z e. U_x e. A ran x <-> E.x e. A z e. ran x)
16	13, 14, 15	3bitr4 183	. 2 |- (z e. ran U. A <-> z e. U_x e. A ran x)
17	16	eqriv 1474	1 |- ran U. A = U_x e. A ran x

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  E.wrex 1646  <.cop 2411  U.cuni 2503  U_ciun 2566  ran crn 3171

This theorem is referenced by:  infxpidmlem6 7557

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-cnv 3186  df-dm 3188  df-rn 3189

Copyright terms: Public domain