Theorem opeldm 4865

Description: Membership of first of an ordered pair in a domain. (Contributed by NM, 30-Jul-1995.)

Hypotheses

Ref	Expression
opeldm.1	⊢ 𝐴 ∈ V
opeldm.2	⊢ 𝐵 ∈ V

Assertion

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

Proof of Theorem opeldm

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeldm.2	. . 3 ⊢ 𝐵 ∈ V
2		opeq2 3805	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
3	2	eleq1d 2262	. . 3 ⊢ (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
4	1, 3	spcev 2855	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐶)
5		opeldm.1	. . 3 ⊢ 𝐴 ∈ V
6	5	eldm2 4860	. 2 ⊢ (𝐴 ∈ dom 𝐶 ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐶)
7	4, 6	sylibr 134	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐴 ∈ dom 𝐶)

Syntax hints:   → wi 4   = wceq 1364  ∃wex 1503   ∈ wcel 2164  Vcvv 2760  ⟨cop 3621  dom cdm 4659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-dm 4669

This theorem is referenced by:  breldm  4866  elreldm  4888  relssres  4980  iss  4988  imadmrn  5015  dfco2a  5166  funssres  5296  funun  5298  iinerm  6661