Theorem rniun 4677

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 2571	. . . 4 ⊢ (∃x ∈ A ∃y⟨y, z⟩ ∈ B ↔ ∃y∃x ∈ A ⟨y, z⟩ ∈ B)
2		vex 2554	. . . . . 6 ⊢ z ∈ V
3	2	elrn2 4519	. . . . 5 ⊢ (z ∈ ran B ↔ ∃y⟨y, z⟩ ∈ B)
4	3	rexbii 2325	. . . 4 ⊢ (∃x ∈ A z ∈ ran B ↔ ∃x ∈ A ∃y⟨y, z⟩ ∈ B)
5		eliun 3652	. . . . 5 ⊢ (⟨y, z⟩ ∈ ∪ x ∈ A B ↔ ∃x ∈ A ⟨y, z⟩ ∈ B)
6	5	exbii 1493	. . . 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 4519	. . 3 ⊢ (z ∈ ran ∪ x ∈ A B ↔ ∃y⟨y, z⟩ ∈ ∪ x ∈ A B)
9		eliun 3652	. . 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 2034	1 ⊢ ran ∪ x ∈ A B = ∪ x ∈ A ran B

Syntax hints:   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301  ⟨cop 3370  ∪ ciun 3648  ran crn 4289

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iun 3650  df-br 3756  df-opab 3810  df-cnv 4296  df-dm 4298  df-rn 4299

This theorem is referenced by:  rnuni  4678  fun11iun  5090