Theorem opeldm 5871

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 4838	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
3	2	eleq1d 2813	. . 3 ⊢ (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
4	1, 3	spcev 3572	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐶)
5		opeldm.1	. . 3 ⊢ 𝐴 ∈ V
6	5	eldm2 5865	. 2 ⊢ (𝐴 ∈ dom 𝐶 ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐶)
7	4, 6	sylibr 234	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐴 ∈ dom 𝐶)

Syntax hints:   → wi 4   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3447  ⟨cop 4595  dom cdm 5638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-dm 5648

This theorem is referenced by:  breldm  5872  elreldm  5899  relssres  5993  iss  6006  imadmrn  6041  dfco2a  6219  relssdmrn  6241  funssres  6560  funun  6562  frxp2  8123  frxp3  8130  frrlem8  8272  frrlem10  8274  tz7.48-1  8411  iiner  8762  r0weon  9965  axdc3lem2  10404  uzrdgfni  13923  imasaddfnlem  17491  imasvscafn  17500  cicsym  17766  gsum2d  19902  noseqrdgfn  28200  cffldtocusgr  29374  cffldtocusgrOLD  29375  dfcnv2  32600  gsumfs2d  32995  bnj1379  34820  iss2  38326  rfovcnvf1od  43993