Theorem dmuni 3308

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

Assertion

Ref	Expression
dmuni	|- dom U. A = U_x e. A dom x

Distinct variable group:   x,A

Proof of Theorem dmuni

Step	Hyp	Ref	Expression
1		eluni 2496	. . . . . 6 |- (<.y, z>. e. U.A <-> E.x(<.y, z>. e. x /\ x e. A))
2	1	exbii 1047	. . . . 5 |- (E.z<.y, z>. e. U.A <-> E.zE.x(<.y, z>. e. x /\ x e. A))
3		excom 1042	. . . . 5 |- (E.zE.x(<.y, z>. e. x /\ x e. A) <-> E.xE.z(<.y, z>. e. x /\ x e. A))
4		ancom 435	. . . . . . 7 |- ((E.z<.y, z>. e. x /\ x e. A) <-> (x e. A /\ E.z<.y, z>. e. x))
5		19.41v 1300	. . . . . . 7 |- (E.z(<.y, z>. e. x /\ x e. A) <-> (E.z<.y, z>. e. x /\ x e. A))
6		visset 1804	. . . . . . . . 9 |- y e. V
7	6	eldm2 3297	. . . . . . . 8 |- (y e. dom x <-> E.z<.y, z>. e. x)
8	7	anbi2i 479	. . . . . . 7 |- ((x e. A /\ y e. dom x) <-> (x e. A /\ E.z<.y, z>. e. x))
9	4, 5, 8	3bitr4 183	. . . . . 6 |- (E.z(<.y, z>. e. x /\ x e. A) <-> (x e. A /\ y e. dom x))
10	9	exbii 1047	. . . . 5 |- (E.xE.z(<.y, z>. e. x /\ x e. A) <-> E.x(x e. A /\ y e. dom x))
11	2, 3, 10	3bitr 177	. . . 4 |- (E.z<.y, z>. e. U.A <-> E.x(x e. A /\ y e. dom x))
12		df-rex 1642	. . . 4 |- (E.x e. A y e. dom x <-> E.x(x e. A /\ y e. dom x))
13	11, 12	bitr4 176	. . 3 |- (E.z<.y, z>. e. U.A <-> E.x e. A y e. dom x)
14	6	eldm2 3297	. . 3 |- (y e. dom U. A <-> E.z<.y, z>. e. U.A)
15		eliun 2560	. . 3 |- (y e. U_x e. A dom x <-> E.x e. A y e. dom x)
16	13, 14, 15	3bitr4 183	. 2 |- (y e. dom U. A <-> y e. U_x e. A dom x)
17	16	eqriv 1467	1 |- dom U. A = U_x e. A dom x

Syntax hints:   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  E.wrex 1638  <.cop 2401  U.cuni 2493  U_ciun 2556  dom cdm 3160

This theorem is referenced by:  tfrlem8 3903  infxpidmlem5 7499  infxpidmlem7 7501

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-rex 1642  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-iun 2558  df-br 2610  df-dm 3178

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