Theorem opeldm 5908

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 4875	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
3	2	eleq1d 2819	. . 3 ⊢ (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
4	1, 3	spcev 3597	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐶)
5		opeldm.1	. . 3 ⊢ 𝐴 ∈ V
6	5	eldm2 5902	. 2 ⊢ (𝐴 ∈ dom 𝐶 ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐶)
7	4, 6	sylibr 233	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐴 ∈ dom 𝐶)

Syntax hints:   → wi 4   = wceq 1542  ∃wex 1782   ∈ wcel 2107  Vcvv 3475  ⟨cop 4635  dom cdm 5677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-dm 5687

This theorem is referenced by:  breldm  5909  elreldm  5935  relssres  6023  iss  6036  imadmrn  6070  dfco2a  6246  relssdmrn  6268  funssres  6593  funun  6595  frxp2  8130  frxp3  8137  frrlem8  8278  frrlem10  8280  tz7.48-1  8443  iiner  8783  r0weon  10007  axdc3lem2  10446  uzrdgfni  13923  imasaddfnlem  17474  imasvscafn  17483  cicsym  17751  gsum2d  19840  cffldtocusgr  28735  dfcnv2  31932  bnj1379  33872  gg-cffldtocusgr  35230  iss2  37261  rfovcnvf1od  42803