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Theorem breldmg 4464
 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.
StepHypRef Expression
1 breq2 3738 . . . . 5 (x = B → (A𝑅xA𝑅B))
21spcegv 2614 . . . 4 (B 𝐷 → (A𝑅Bx A𝑅x))
32imp 115 . . 3 ((B 𝐷 A𝑅B) → x A𝑅x)
433adant1 908 . 2 ((A 𝐶 B 𝐷 A𝑅B) → x A𝑅x)
5 eldmg 4453 . . 3 (A 𝐶 → (A dom 𝑅x A𝑅x))
653ad2ant1 911 . 2 ((A 𝐶 B 𝐷 A𝑅B) → (A dom 𝑅x A𝑅x))
74, 6mpbird 156 1 ((A 𝐶 B 𝐷 A𝑅B) → A dom 𝑅)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∧ w3a 871  ∃wex 1358   ∈ wcel 1370   class class class wbr 3734  dom cdm 4268 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-dm 4278 This theorem is referenced by:  brelrng  4488  releldm  4492  brtposg  5787
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