Theorem elreldm 4830

Description: The first member of an ordered pair in a relation belongs to the domain of the relation. (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 4611	. . . . 5 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
2		ssel 3136	. . . . 5 ⊢ (𝐴 ⊆ (V × V) → (𝐵 ∈ 𝐴 → 𝐵 ∈ (V × V)))
3	1, 2	sylbi 120	. . . 4 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ∈ (V × V)))
4		elvv 4666	. . . 4 ⊢ (𝐵 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
5	3, 4	syl6ib 160	. . 3 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → ∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩))
6		eleq1 2229	. . . . . 6 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵 ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
7		vex 2729	. . . . . . 7 ⊢ 𝑥 ∈ V
8		vex 2729	. . . . . . 7 ⊢ 𝑦 ∈ V
9	7, 8	opeldm 4807	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
10	6, 9	syl6bi 162	. . . . 5 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵 ∈ 𝐴 → 𝑥 ∈ dom 𝐴))
11		inteq 3827	. . . . . . . 8 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → ∩ 𝐵 = ∩ ⟨𝑥, 𝑦⟩)
12	11	inteqd 3829	. . . . . . 7 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → ∩ ∩ 𝐵 = ∩ ∩ ⟨𝑥, 𝑦⟩)
13	7, 8	op1stb 4456	. . . . . . 7 ⊢ ∩ ∩ ⟨𝑥, 𝑦⟩ = 𝑥
14	12, 13	eqtrdi 2215	. . . . . 6 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → ∩ ∩ 𝐵 = 𝑥)
15	14	eleq1d 2235	. . . . 5 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → (∩ ∩ 𝐵 ∈ dom 𝐴 ↔ 𝑥 ∈ dom 𝐴))
16	10, 15	sylibrd 168	. . . 4 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵 ∈ 𝐴 → ∩ ∩ 𝐵 ∈ dom 𝐴))
17	16	exlimivv 1884	. . 3 ⊢ (∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵 ∈ 𝐴 → ∩ ∩ 𝐵 ∈ dom 𝐴))
18	5, 17	syli 37	. 2 ⊢ (Rel 𝐴 → (𝐵 ∈ 𝐴 → ∩ ∩ 𝐵 ∈ dom 𝐴))
19	18	imp 123	1 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → ∩ ∩ 𝐵 ∈ dom 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343  ∃wex 1480   ∈ wcel 2136  Vcvv 2726   ⊆ wss 3116  ⟨cop 3579  ∩ cint 3824   × cxp 4602  dom cdm 4604  Rel wrel 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-int 3825  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-dm 4614

This theorem is referenced by:  1stdm  6150  fundmen  6772