Theorem rniun 4661

Description: The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

Assertion

Ref	Expression
rniun	⊢ ran ∪ x ∈ A B = ∪ x ∈ A ran B

Proof of Theorem rniun

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

Step	Hyp	Ref	Expression
1		rexcom4 2554	. . . 4 ⊢ (∃x ∈ A ∃y⟨y, z⟩ ∈ B ↔ ∃y∃x ∈ A ⟨y, z⟩ ∈ B)
2		vex 2538	. . . . . 6 ⊢ z ∈ V
3	2	elrn2 4503	. . . . 5 ⊢ (z ∈ ran B ↔ ∃y⟨y, z⟩ ∈ B)
4	3	rexbii 2309	. . . 4 ⊢ (∃x ∈ A z ∈ ran B ↔ ∃x ∈ A ∃y⟨y, z⟩ ∈ B)
5		eliun 3635	. . . . 5 ⊢ (⟨y, z⟩ ∈ ∪ x ∈ A B ↔ ∃x ∈ A ⟨y, z⟩ ∈ B)
6	5	exbii 1478	. . . 4 ⊢ (∃y⟨y, z⟩ ∈ ∪ x ∈ A B ↔ ∃y∃x ∈ A ⟨y, z⟩ ∈ B)
7	1, 4, 6	3bitr4ri 202	. . 3 ⊢ (∃y⟨y, z⟩ ∈ ∪ x ∈ A B ↔ ∃x ∈ A z ∈ ran B)
8	2	elrn2 4503	. . 3 ⊢ (z ∈ ran ∪ x ∈ A B ↔ ∃y⟨y, z⟩ ∈ ∪ x ∈ A B)
9		eliun 3635	. . 3 ⊢ (z ∈ ∪ x ∈ A ran B ↔ ∃x ∈ A z ∈ ran B)
10	7, 8, 9	3bitr4i 201	. 2 ⊢ (z ∈ ran ∪ x ∈ A B ↔ z ∈ ∪ x ∈ A ran B)
11	10	eqriv 2019	1 ⊢ ran ∪ x ∈ A B = ∪ x ∈ A ran B

Syntax hints:   = wceq 1228  ∃wex 1362   ∈ wcel 1374  ∃wrex 2285  ⟨cop 3353  ∪ ciun 3631  ran crn 4273

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-iun 3633  df-br 3739  df-opab 3793  df-cnv 4280  df-dm 4282  df-rn 4283

This theorem is referenced by:  rnuni  4662  fun11iun  5072