Theorem dmiun 4544

Description: The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.)

Assertion

Ref	Expression
dmiun	⊢ dom ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 dom 𝐵

Proof of Theorem dmiun

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

Step	Hyp	Ref	Expression
1		rexcom4 2577	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐵 ↔ ∃𝑧∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ 𝐵)
2		vex 2560	. . . . . 6 ⊢ 𝑦 ∈ V
3	2	eldm2 4533	. . . . 5 ⊢ (𝑦 ∈ dom 𝐵 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐵)
4	3	rexbii 2331	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐵)
5		eliun 3661	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ 𝐵)
6	5	exbii 1496	. . . 4 ⊢ (∃𝑧⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑧∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ 𝐵)
7	1, 4, 6	3bitr4ri 202	. . 3 ⊢ (∃𝑧⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝐵)
8	2	eldm2 4533	. . 3 ⊢ (𝑦 ∈ dom ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
9		eliun 3661	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝐵)
10	7, 8, 9	3bitr4i 201	. 2 ⊢ (𝑦 ∈ dom ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝐵)
11	10	eqriv 2037	1 ⊢ dom ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 dom 𝐵

Syntax hints:   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307  ⟨cop 3378  ∪ ciun 3657  dom cdm 4345

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-iun 3659  df-br 3765  df-dm 4355

This theorem is referenced by: (None)