ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eldm2g GIF version

Theorem eldm2g 4558
Description: Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
eldm2g (𝐴𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐵))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem eldm2g
StepHypRef Expression
1 eldmg 4557 . 2 (𝐴𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
2 df-br 3792 . . 3 (𝐴𝐵𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐵)
32exbii 1512 . 2 (∃𝑦 𝐴𝐵𝑦 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐵)
41, 3syl6bb 189 1 (𝐴𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wex 1397  wcel 1409  cop 3405   class class class wbr 3791  dom cdm 4372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-dm 4382
This theorem is referenced by:  eldm2  4560  opeldmg  4567  dmfco  5268  releldm2  5838  tfrlem9  5965  climcau  10089
  Copyright terms: Public domain W3C validator