Theorem breldmg 4559

Description: Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)

Assertion

Ref	Expression
breldmg	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Proof of Theorem breldmg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 3789	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵))
2	1	spcegv 2686	. . . 4 ⊢ (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 → ∃𝑥 𝐴𝑅𝑥))
3	2	imp 122	. . 3 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → ∃𝑥 𝐴𝑅𝑥)
4	3	3adant1 956	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → ∃𝑥 𝐴𝑅𝑥)
5		eldmg 4548	. . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
6	5	3ad2ant1 959	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
7	4, 6	mpbird 165	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ↔ wb 103   ∧ w3a 919  ∃wex 1421   ∈ wcel 1433   class class class wbr 3785  dom cdm 4363

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-dm 4373

This theorem is referenced by:  brelrng  4583  releldm  4587  brtposg  5892  shftfvalg  9706  shftfval  9709