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Theorem opeldmg 4814
Description: Membership of first of an ordered pair in a domain. (Contributed by Jim Kingdon, 9-Jul-2019.)
Assertion
Ref Expression
opeldmg ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 ∈ dom 𝐶))

Proof of Theorem opeldmg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 opeq2 3764 . . . . 5 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
21eleq1d 2239 . . . 4 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
32spcegv 2818 . . 3 (𝐵𝑊 → (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶))
43adantl 275 . 2 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶))
5 eldm2g 4805 . . 3 (𝐴𝑉 → (𝐴 ∈ dom 𝐶 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶))
65adantr 274 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ dom 𝐶 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶))
74, 6sylibrd 168 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 ∈ dom 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wex 1485  wcel 2141  cop 3584  dom cdm 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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-dm 4619
This theorem is referenced by:  tfr0dm  6299  tfrlemi14d  6310  tfr1onlemres  6326  tfrcllemres  6339  fnfi  6911  frecuzrdgtcl  10357  frecuzrdgdomlem  10362  hashennn  10703
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