Theorem opeldm 3309

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 2484	. . . . 5 ⊢ (y = B → ⟨A, y⟩ = ⟨A, B⟩)
2	1	eleq1d 1537	. . . 4 ⊢ (y = B → (⟨A, y⟩ ∈ C ↔ ⟨A, B⟩ ∈ C))
3	2	cla4egv 1859	. . 3 ⊢ (B ∈ V → (⟨A, B⟩ ∈ C → ∃y⟨A, y⟩ ∈ C))
4		opeldm.1	. . . 4 ⊢ A ∈ V
5	4	eldm2 3303	. . 3 ⊢ (A ∈ dom C ↔ ∃y⟨A, y⟩ ∈ C)
6	3, 5	syl6ibr 213	. 2 ⊢ (B ∈ V → (⟨A, B⟩ ∈ C → A ∈ dom C))
7		opprc2 2495	. . . 4 ⊢ (¬ B ∈ V → ⟨A, B⟩ = ⟨A, A⟩)
8	7	eleq1d 1537	. . 3 ⊢ (¬ B ∈ V → (⟨A, B⟩ ∈ C ↔ ⟨A, A⟩ ∈ C))
9		opeq2 2484	. . . . . 6 ⊢ (y = A → ⟨A, y⟩ = ⟨A, A⟩)
10	9	eleq1d 1537	. . . . 5 ⊢ (y = A → (⟨A, y⟩ ∈ C ↔ ⟨A, A⟩ ∈ C))
11	4, 10	cla4ev 1865	. . . 4 ⊢ (⟨A, A⟩ ∈ C → ∃y⟨A, y⟩ ∈ C)
12	11, 5	sylibr 200	. . 3 ⊢ (⟨A, A⟩ ∈ C → A ∈ dom C)
13	8, 12	syl6bi 214	. 2 ⊢ (¬ B ∈ V → (⟨A, B⟩ ∈ C → A ∈ dom C))
14	6, 13	pm2.61i 126	1 ⊢ (⟨A, B⟩ ∈ C → A ∈ dom C)

Syntax hints:  ¬ wn 2   → wi 3   = wceq 954   ∈ wcel 956  ∃wex 978  Vcvv 1807  ⟨cop 2407  dom cdm 3165

This theorem is referenced by:  breldm 3310  elreldm 3333  relssres 3384  imadmrn 3406  funssres 3544  funun 3546  fnrnfv 3750  eqfnfv 3788  tz7.48-1 3947  ecopoprdm 4299

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-nul 2277  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-dm 3183

Copyright terms: Public domain