Theorem elreldm 5945

Description: The first member of an ordered pair in a relation belongs to the domain of the relation (see op1stb 5475). (Contributed by NM, 28-Jul-2004.)

Assertion

Ref	Expression
elreldm	⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → ∩ ∩ 𝐵 ∈ dom 𝐴)

Proof of Theorem elreldm

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

Step	Hyp	Ref	Expression
1		df-rel 5691	. . . . 5 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
2		ssel 3976	. . . . 5 ⊢ (𝐴 ⊆ (V × V) → (𝐵 ∈ 𝐴 → 𝐵 ∈ (V × V)))
3	1, 2	sylbi 217	. . . 4 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ∈ (V × V)))
4		elvv 5759	. . . 4 ⊢ (𝐵 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
5	3, 4	imbitrdi 251	. . 3 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → ∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩))
6		eleq1 2828	. . . . . 6 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵 ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
7		vex 3483	. . . . . . 7 ⊢ 𝑥 ∈ V
8		vex 3483	. . . . . . 7 ⊢ 𝑦 ∈ V
9	7, 8	opeldm 5917	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
10	6, 9	biimtrdi 253	. . . . 5 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵 ∈ 𝐴 → 𝑥 ∈ dom 𝐴))
11		inteq 4948	. . . . . . . 8 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → ∩ 𝐵 = ∩ ⟨𝑥, 𝑦⟩)
12	11	inteqd 4950	. . . . . . 7 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → ∩ ∩ 𝐵 = ∩ ∩ ⟨𝑥, 𝑦⟩)
13	7, 8	op1stb 5475	. . . . . . 7 ⊢ ∩ ∩ ⟨𝑥, 𝑦⟩ = 𝑥
14	12, 13	eqtrdi 2792	. . . . . 6 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → ∩ ∩ 𝐵 = 𝑥)
15	14	eleq1d 2825	. . . . 5 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → (∩ ∩ 𝐵 ∈ dom 𝐴 ↔ 𝑥 ∈ dom 𝐴))
16	10, 15	sylibrd 259	. . . 4 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵 ∈ 𝐴 → ∩ ∩ 𝐵 ∈ dom 𝐴))
17	16	exlimivv 1931	. . 3 ⊢ (∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵 ∈ 𝐴 → ∩ ∩ 𝐵 ∈ dom 𝐴))
18	5, 17	syli 39	. 2 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → ∩ ∩ 𝐵 ∈ dom 𝐴))
19	18	imp 406	1 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → ∩ ∩ 𝐵 ∈ dom 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  Vcvv 3479   ⊆ wss 3950  ⟨cop 4631  ∩ cint 4945   × cxp 5682  dom cdm 5684  Rel wrel 5689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-int 4946  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-dm 5694

This theorem is referenced by:  1stdm  8066  fundmen  9072