Theorem breldmg 4935

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 4090	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵))
2	1	spcegv 2892	. . . 4 ⊢ (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 → ∃𝑥 𝐴𝑅𝑥))
3	2	imp 124	. . 3 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → ∃𝑥 𝐴𝑅𝑥)
4	3	3adant1 1039	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → ∃𝑥 𝐴𝑅𝑥)
5		eldmg 4924	. . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
6	5	3ad2ant1 1042	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
7	4, 6	mpbird 167	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 1002  ∃wex 1538   ∈ wcel 2200   class class class wbr 4086  dom cdm 4723

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-dm 4733

This theorem is referenced by:  brelrng  4961  releldm  4965  brtposg  6415  shftfvalg  11369  shftfval  11372  geolim2  12063  geoisum1c  12071  ntrivcvgap  12099  eftlub  12241  eflegeo  12252  dvcj  15423  dvrecap  15427  dvef  15441  trilpolemisumle  16578  trilpolemeq1  16580