Theorem elreldm 5874

Description: The first member of an ordered pair in a relation belongs to the domain of the relation (see op1stb 5409). (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 5621	. . . . 5 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
2		ssel 3923	. . . . 5 ⊢ (𝐴 ⊆ (V × V) → (𝐵 ∈ 𝐴 → 𝐵 ∈ (V × V)))
3	1, 2	sylbi 217	. . . 4 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ∈ (V × V)))
4		elvv 5689	. . . 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 2819	. . . . . 6 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵 ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
7		vex 3440	. . . . . . 7 ⊢ 𝑥 ∈ V
8		vex 3440	. . . . . . 7 ⊢ 𝑦 ∈ V
9	7, 8	opeldm 5846	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
10	6, 9	biimtrdi 253	. . . . 5 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵 ∈ 𝐴 → 𝑥 ∈ dom 𝐴))
11		inteq 4898	. . . . . . . 8 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → ∩ 𝐵 = ∩ ⟨𝑥, 𝑦⟩)
12	11	inteqd 4900	. . . . . . 7 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → ∩ ∩ 𝐵 = ∩ ∩ ⟨𝑥, 𝑦⟩)
13	7, 8	op1stb 5409	. . . . . . 7 ⊢ ∩ ∩ ⟨𝑥, 𝑦⟩ = 𝑥
14	12, 13	eqtrdi 2782	. . . . . 6 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → ∩ ∩ 𝐵 = 𝑥)
15	14	eleq1d 2816	. . . . 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 1933	. . 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 1541  ∃wex 1780   ∈ wcel 2111  Vcvv 3436   ⊆ wss 3897  ⟨cop 4579  ∩ cint 4895   × cxp 5612  dom cdm 5614  Rel wrel 5619

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-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

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-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-int 4896  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-dm 5624

This theorem is referenced by:  1stdm  7972  fundmen  8953