Theorem opeldm 2534

Description: Membership of first of an ordered pair in a domain.

Hypothesis

Ref	Expression
opeldm.1	⊢ A ∈ V

Assertion

Ref	Expression
opeldm	⊢ (⟨A, B⟩ ∈ C → A ∈ dom C)

Proof of Theorem opeldm

Step	Hyp	Ref	Expression
1		opeq2 1877	. . . . 5 ⊢ (y = B → ⟨A, y⟩ = ⟨A, B⟩)
2	1	eleq1d 1155	. . . 4 ⊢ (y = B → (⟨A, y⟩ ∈ C ↔ ⟨A, B⟩ ∈ C))
3	2	cla4egv 1397	. . 3 ⊢ (B ∈ V → (⟨A, B⟩ ∈ C → ∃y⟨A, y⟩ ∈ C))
4		opeldm.1	. . . 4 ⊢ A ∈ V
5	4	eldm2 2528	. . 3 ⊢ (A ∈ dom C ↔ ∃y⟨A, y⟩ ∈ C)
6	3, 5	syl6ibr 186	. 2 ⊢ (B ∈ V → (⟨A, B⟩ ∈ C → A ∈ dom C))
7		opprc2 1907	. . . 4 ⊢ (¬ B ∈ V → ⟨A, B⟩ = ⟨A, A⟩)
8	7	eleq1d 1155	. . 3 ⊢ (¬ B ∈ V → (⟨A, B⟩ ∈ C ↔ ⟨A, A⟩ ∈ C))
9		opeq2 1877	. . . . . 6 ⊢ (y = A → ⟨A, y⟩ = ⟨A, A⟩)
10	9	eleq1d 1155	. . . . 5 ⊢ (y = A → (⟨A, y⟩ ∈ C ↔ ⟨A, A⟩ ∈ C))
11	4, 10	cla4ev 1401	. . . 4 ⊢ (⟨A, A⟩ ∈ C → ∃y⟨A, y⟩ ∈ C)
12	11, 5	sylibr 175	. . 3 ⊢ (⟨A, A⟩ ∈ C → A ∈ dom C)
13	8, 12	syl6bi 187	. 2 ⊢ (¬ B ∈ V → (⟨A, B⟩ ∈ C → A ∈ dom C))
14	6, 13	pm2.61i 110	1 ⊢ (⟨A, B⟩ ∈ C → A ∈ dom C)

Syntax hints:  ¬ wn 1   → wi 2  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810  dom cdm 2410

This theorem is referenced by:  breldm 2535  elreldm 2554  relssres 2596  imadmrn 2610  funssres 2698  funun 2700  fniunfv 2860  cleqfv 2880  tz7.48-1 2994  ecopoprdm 3245

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-16 922  ax-17 925  ax-ext 1074

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-v 1349  df-dif 1489  df-un 1490  df-nul 1708  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-dm 2428

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