Theorem elreldm 5899

Description: The first member of an ordered pair in a relation belongs to the domain of the relation (see op1stb 5431). (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 5645	. . . . 5 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
2		ssel 3940	. . . . 5 ⊢ (𝐴 ⊆ (V × V) → (𝐵 ∈ 𝐴 → 𝐵 ∈ (V × V)))
3	1, 2	sylbi 217	. . . 4 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ∈ (V × V)))
4		elvv 5713	. . . 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 2816	. . . . . 6 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵 ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
7		vex 3451	. . . . . . 7 ⊢ 𝑥 ∈ V
8		vex 3451	. . . . . . 7 ⊢ 𝑦 ∈ V
9	7, 8	opeldm 5871	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
10	6, 9	biimtrdi 253	. . . . 5 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵 ∈ 𝐴 → 𝑥 ∈ dom 𝐴))
11		inteq 4913	. . . . . . . 8 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → ∩ 𝐵 = ∩ ⟨𝑥, 𝑦⟩)
12	11	inteqd 4915	. . . . . . 7 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → ∩ ∩ 𝐵 = ∩ ∩ ⟨𝑥, 𝑦⟩)
13	7, 8	op1stb 5431	. . . . . . 7 ⊢ ∩ ∩ ⟨𝑥, 𝑦⟩ = 𝑥
14	12, 13	eqtrdi 2780	. . . . . 6 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → ∩ ∩ 𝐵 = 𝑥)
15	14	eleq1d 2813	. . . . 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 1932	. . 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 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3447   ⊆ wss 3914  ⟨cop 4595  ∩ cint 4910   × cxp 5636  dom cdm 5638  Rel wrel 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-int 4911  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-dm 5648

This theorem is referenced by:  1stdm  8019  fundmen  9002