Theorem rnuni 5039

Description: The range of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by set.mm contributors, 17-Mar-2004.)

Assertion

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

Distinct variable group:   x,A

Proof of Theorem rnuni

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 3895	. . . . . 6 ⊢ (⟨y, z⟩ ∈ ∪A ↔ ∃x(⟨y, z⟩ ∈ x ∧ x ∈ A))
2	1	exbii 1582	. . . . 5 ⊢ (∃y⟨y, z⟩ ∈ ∪A ↔ ∃y∃x(⟨y, z⟩ ∈ x ∧ x ∈ A))
3		excom 1741	. . . . 5 ⊢ (∃y∃x(⟨y, z⟩ ∈ x ∧ x ∈ A) ↔ ∃x∃y(⟨y, z⟩ ∈ x ∧ x ∈ A))
4		elrn2 4898	. . . . . . . 8 ⊢ (z ∈ ran x ↔ ∃y⟨y, z⟩ ∈ x)
5	4	anbi1i 676	. . . . . . 7 ⊢ ((z ∈ ran x ∧ x ∈ A) ↔ (∃y⟨y, z⟩ ∈ x ∧ x ∈ A))
6		ancom 437	. . . . . . 7 ⊢ ((x ∈ A ∧ z ∈ ran x) ↔ (z ∈ ran x ∧ x ∈ A))
7		19.41v 1901	. . . . . . 7 ⊢ (∃y(⟨y, z⟩ ∈ x ∧ x ∈ A) ↔ (∃y⟨y, z⟩ ∈ x ∧ x ∈ A))
8	5, 6, 7	3bitr4ri 269	. . . . . 6 ⊢ (∃y(⟨y, z⟩ ∈ x ∧ x ∈ A) ↔ (x ∈ A ∧ z ∈ ran x))
9	8	exbii 1582	. . . . 5 ⊢ (∃x∃y(⟨y, z⟩ ∈ x ∧ x ∈ A) ↔ ∃x(x ∈ A ∧ z ∈ ran x))
10	2, 3, 9	3bitri 262	. . . 4 ⊢ (∃y⟨y, z⟩ ∈ ∪A ↔ ∃x(x ∈ A ∧ z ∈ ran x))
11		df-rex 2621	. . . 4 ⊢ (∃x ∈ A z ∈ ran x ↔ ∃x(x ∈ A ∧ z ∈ ran x))
12	10, 11	bitr4i 243	. . 3 ⊢ (∃y⟨y, z⟩ ∈ ∪A ↔ ∃x ∈ A z ∈ ran x)
13		elrn2 4898	. . 3 ⊢ (z ∈ ran ∪A ↔ ∃y⟨y, z⟩ ∈ ∪A)
14		eliun 3974	. . 3 ⊢ (z ∈ ∪x ∈ A ran x ↔ ∃x ∈ A z ∈ ran x)
15	12, 13, 14	3bitr4i 268	. 2 ⊢ (z ∈ ran ∪A ↔ z ∈ ∪x ∈ A ran x)
16	15	eqriv 2350	1 ⊢ ran ∪A = ∪x ∈ A ran x

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  ∪cuni 3892  ∪ciun 3970  ⟨cop 4562  ran crn 4774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-br 4641  df-ima 4728  df-rn 4787

This theorem is referenced by: (None)