Theorem rnuni 2646

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

Assertion

Ref	Expression
rnuni	⊢ ran ∪A = ∪x ∈ A ran x

Distinct variable group(s):   x,A

Proof of Theorem rnuni

Step	Hyp	Ref	Expression
1		eluni 1922	. . . . . 6 ⊢ (⟨y, z⟩ ∈ ∪A ↔ ∃x(⟨y, z⟩ ∈ x ∧ x ∈ A))
2	1	biex 733	. . . . 5 ⊢ (∃y⟨y, z⟩ ∈ ∪A ↔ ∃y∃x(⟨y, z⟩ ∈ x ∧ x ∈ A))
3		excom 728	. . . . 5 ⊢ (∃y∃x(⟨y, z⟩ ∈ x ∧ x ∈ A) ↔ ∃x∃y(⟨y, z⟩ ∈ x ∧ x ∈ A))
4		ancom 333	. . . . . . 7 ⊢ ((∃y⟨y, z⟩ ∈ x ∧ x ∈ A) ↔ (x ∈ A ∧ ∃y⟨y, z⟩ ∈ x))
5		19.41v 963	. . . . . . 7 ⊢ (∃y(⟨y, z⟩ ∈ x ∧ x ∈ A) ↔ (∃y⟨y, z⟩ ∈ x ∧ x ∈ A))
6		visset 1350	. . . . . . . . 9 ⊢ z ∈ V
7	6	elrn 2562	. . . . . . . 8 ⊢ (z ∈ ran x ↔ ∃y⟨y, z⟩ ∈ x)
8	7	anbi2i 367	. . . . . . 7 ⊢ ((x ∈ A ∧ z ∈ ran x) ↔ (x ∈ A ∧ ∃y⟨y, z⟩ ∈ x))
9	4, 5, 8	3bitr4 158	. . . . . 6 ⊢ (∃y(⟨y, z⟩ ∈ x ∧ x ∈ A) ↔ (x ∈ A ∧ z ∈ ran x))
10	9	biex 733	. . . . 5 ⊢ (∃x∃y(⟨y, z⟩ ∈ x ∧ x ∈ A) ↔ ∃x(x ∈ A ∧ z ∈ ran x))
11	2, 3, 10	3bitr 155	. . . 4 ⊢ (∃y⟨y, z⟩ ∈ ∪A ↔ ∃x(x ∈ A ∧ z ∈ ran x))
12		df-rex 1206	. . . 4 ⊢ (∃x ∈ A z ∈ ran x ↔ ∃x(x ∈ A ∧ z ∈ ran x))
13	11, 12	bitr4 154	. . 3 ⊢ (∃y⟨y, z⟩ ∈ ∪A ↔ ∃x ∈ A z ∈ ran x)
14	6	elrn 2562	. . 3 ⊢ (z ∈ ran ∪A ↔ ∃y⟨y, z⟩ ∈ ∪A)
15		eliun 1998	. . 3 ⊢ (z ∈ ∪x ∈ A ran x ↔ ∃x ∈ A z ∈ ran x)
16	13, 14, 15	3bitr4 158	. 2 ⊢ (z ∈ ran ∪A ↔ z ∈ ∪x ∈ A ran x)
17	16	cleqri 1101	1 ⊢ ran ∪A = ∪x ∈ A ran x

Syntax hints:   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  ⟨cop 1810  ∪cuni 1919  ∪ciun 1994  ran crn 2411

This theorem is referenced by:  infxpidmlem6 4938

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-iun 1996  df-br 2063  df-opab 2098  df-cnv 2426  df-dm 2428  df-rn 2429

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