Theorem rniun 6143

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 3298	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦⟨𝑦, 𝑧⟩ ∈ 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ 𝐵)
2		vex 3467	. . . . . 6 ⊢ 𝑧 ∈ V
3	2	elrn2 5880	. . . . 5 ⊢ (𝑧 ∈ ran 𝐵 ↔ ∃𝑦⟨𝑦, 𝑧⟩ ∈ 𝐵)
4	3	rexbii 3118	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦⟨𝑦, 𝑧⟩ ∈ 𝐵)
5		eliun 4961	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ 𝐵)
6	5	exbii 1875	. . . 4 ⊢ (∃𝑦⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ 𝐵)
7	1, 4, 6	3bitr4ri 307	. . 3 ⊢ (∃𝑦⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵)
8	2	elrn2 5880	. . 3 ⊢ (𝑧 ∈ ran ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑦⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
9		eliun 4961	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ran 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵)
10	7, 8, 9	3bitr4i 306	. 2 ⊢ (𝑧 ∈ ran ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ran 𝐵)
11	10	eqriv 2766	1 ⊢ ran ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ran 𝐵

Syntax hints:   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∃wrex 3095  ⟨cop 4597  ∪ ciun 4957  ran crn 5660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-iun 4959  df-br 5111  df-opab 5175  df-cnv 5667  df-dm 5669  df-rn 5670

This theorem is referenced by:  rnuni  6144  fiun  7936  f1iun  7937  cnextf  24188  iunrelexp0  44313