Theorem opeldm 5862

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 4817	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
3	2	eleq1d 2821	. . 3 ⊢ (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
4	1, 3	spcev 3548	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐶)
5		opeldm.1	. . 3 ⊢ 𝐴 ∈ V
6	5	eldm2 5856	. 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 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3429  ⟨cop 4573  dom cdm 5631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-dm 5641

This theorem is referenced by:  breldm  5863  elreldm  5890  relssres  5987  iss  6000  imadmrn  6035  dfco2a  6210  relssdmrn  6233  funssres  6542  funun  6544  frxp2  8094  frxp3  8101  frrlem8  8243  frrlem10  8245  tz7.48-1  8382  iiner  8736  r0weon  9934  axdc3lem2  10373  uzrdgfni  13920  imasaddfnlem  17492  imasvscafn  17501  cicsym  17771  gsum2d  19947  noseqrdgfn  28298  cffldtocusgr  29516  dfcnv2  32748  gsumfs2d  33122  bnj1379  34972  iss2  38665  rfovcnvf1od  44431