Theorem dmuni

Description: The domain of a union. Part of Exercise 8 of Enderton p. 41. (Contributed by NM, 3-Feb-2004)

	Ref	Expression
Assertion	dmuni	⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥

Proof

Step	Hyp	Ref	Expression
1		excom	⊢ ( ∃ 𝑧 ∃ 𝑥 ( ⟨ 𝑦 , 𝑧 ⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) ↔ ∃ 𝑥 ∃ 𝑧 ( ⟨ 𝑦 , 𝑧 ⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) )
2		ancom	⊢ ( ( ∃ 𝑧 ⟨ 𝑦 , 𝑧 ⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑧 ⟨ 𝑦 , 𝑧 ⟩ ∈ 𝑥 ) )
3		19.41v	⊢ ( ∃ 𝑧 ( ⟨ 𝑦 , 𝑧 ⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) ↔ ( ∃ 𝑧 ⟨ 𝑦 , 𝑧 ⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) )
4		vex	⊢ 𝑦 ∈ V
5	4	eldm2	⊢ ( 𝑦 ∈ dom 𝑥 ↔ ∃ 𝑧 ⟨ 𝑦 , 𝑧 ⟩ ∈ 𝑥 )
6	5	anbi2i	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑧 ⟨ 𝑦 , 𝑧 ⟩ ∈ 𝑥 ) )
7	2 3 6	3bitr4i	⊢ ( ∃ 𝑧 ( ⟨ 𝑦 , 𝑧 ⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥 ) )
8	7	exbii	⊢ ( ∃ 𝑥 ∃ 𝑧 ( ⟨ 𝑦 , 𝑧 ⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥 ) )
9	1 8	bitri	⊢ ( ∃ 𝑧 ∃ 𝑥 ( ⟨ 𝑦 , 𝑧 ⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥 ) )
10		eluni	⊢ ( ⟨ 𝑦 , 𝑧 ⟩ ∈ ∪ 𝐴 ↔ ∃ 𝑥 ( ⟨ 𝑦 , 𝑧 ⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) )
11	10	exbii	⊢ ( ∃ 𝑧 ⟨ 𝑦 , 𝑧 ⟩ ∈ ∪ 𝐴 ↔ ∃ 𝑧 ∃ 𝑥 ( ⟨ 𝑦 , 𝑧 ⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) )
12		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥 ) )
13	9 11 12	3bitr4i	⊢ ( ∃ 𝑧 ⟨ 𝑦 , 𝑧 ⟩ ∈ ∪ 𝐴 ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥 )
14	4	eldm2	⊢ ( 𝑦 ∈ dom ∪ 𝐴 ↔ ∃ 𝑧 ⟨ 𝑦 , 𝑧 ⟩ ∈ ∪ 𝐴 )
15		eliun	⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝑥 ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥 )
16	13 14 15	3bitr4i	⊢ ( 𝑦 ∈ dom ∪ 𝐴 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝑥 )
17	16	eqriv	⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥