Theorem elreldm 2554

Description: The first member of an ordered pair in a relation belongs to the domain of the relation.

Assertion

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

Proof of Theorem elreldm

Step	Hyp	Ref	Expression
1		df-rel 2425	. . . . 5 ⊢ (Rel A ↔ A ⊆ (V × V))
2		ssel 1502	. . . . 5 ⊢ (A ⊆ (V × V) → (B ∈ A → B ∈ (V × V)))
3	1, 2	sylbi 174	. . . 4 ⊢ (Rel A → (B ∈ A → B ∈ (V × V)))
4		elvv 2464	. . . 4 ⊢ (B ∈ (V × V) ↔ ∃x∃y B = ⟨x, y⟩)
5	3, 4	syl6ib 185	. . 3 ⊢ (Rel A → (B ∈ A → ∃x∃y B = ⟨x, y⟩))
6		eleq1 1149	. . . . . 6 ⊢ (B = ⟨x, y⟩ → (B ∈ A ↔ ⟨x, y⟩ ∈ A))
7		visset 1350	. . . . . . 7 ⊢ x ∈ V
8	7	opeldm 2534	. . . . . 6 ⊢ (⟨x, y⟩ ∈ A → x ∈ dom A)
9	6, 8	syl6bi 187	. . . . 5 ⊢ (B = ⟨x, y⟩ → (B ∈ A → x ∈ dom A))
10		inteq 1968	. . . . . . . 8 ⊢ (B = ⟨x, y⟩ → ∩B = ∩⟨x, y⟩)
11	10	inteqd 1970	. . . . . . 7 ⊢ (B = ⟨x, y⟩ → ∩∩B = ∩∩⟨x, y⟩)
12	7	op1stb 1992	. . . . . . 7 ⊢ ∩∩⟨x, y⟩ = x
13	11, 12	syl6eq 1140	. . . . . 6 ⊢ (B = ⟨x, y⟩ → ∩∩B = x)
14	13	eleq1d 1155	. . . . 5 ⊢ (B = ⟨x, y⟩ → (∩∩B ∈ dom A ↔ x ∈ dom A))
15	9, 14	sylibrd 179	. . . 4 ⊢ (B = ⟨x, y⟩ → (B ∈ A → ∩∩B ∈ dom A))
16	15	19.23aivv 953	. . 3 ⊢ (∃x∃y B = ⟨x, y⟩ → (B ∈ A → ∩∩B ∈ dom A))
17	5, 16	syli 52	. 2 ⊢ (Rel A → (B ∈ A → ∩∩B ∈ dom A))
18	17	imp 277	1 ⊢ ((Rel A ∧ B ∈ A) → ∩∩B ∈ dom A)

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ⟨cop 1810  ∩cint 1965   × cxp 2408  dom cdm 2410  Rel wrel 2415

This theorem is referenced by:  fundmen 3333

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-int 1966  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-dm 2428

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