Theorem rniun 6094

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 3259	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦⟨𝑦, 𝑧⟩ ∈ 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ 𝐵)
2		vex 3440	. . . . . 6 ⊢ 𝑧 ∈ V
3	2	elrn2 5832	. . . . 5 ⊢ (𝑧 ∈ ran 𝐵 ↔ ∃𝑦⟨𝑦, 𝑧⟩ ∈ 𝐵)
4	3	rexbii 3079	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦⟨𝑦, 𝑧⟩ ∈ 𝐵)
5		eliun 4945	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ 𝐵)
6	5	exbii 1849	. . . 4 ⊢ (∃𝑦⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ 𝐵)
7	1, 4, 6	3bitr4ri 304	. . 3 ⊢ (∃𝑦⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵)
8	2	elrn2 5832	. . 3 ⊢ (𝑧 ∈ ran ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑦⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
9		eliun 4945	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ran 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵)
10	7, 8, 9	3bitr4i 303	. 2 ⊢ (𝑧 ∈ ran ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ran 𝐵)
11	10	eqriv 2728	1 ⊢ ran ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ran 𝐵

Syntax hints:   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∃wrex 3056  ⟨cop 4582  ∪ ciun 4941  ran crn 5617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-11 2160  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-iun 4943  df-br 5092  df-opab 5154  df-cnv 5624  df-dm 5626  df-rn 5627

This theorem is referenced by:  rnuni  6095  fiun  7875  f1iun  7876  cnextf  23982  iunrelexp0  43741