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Theorem eldm2g 4474
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 (A 𝑉 → (A dom ByA, y B))
Distinct variable groups:   y,A   y,B
Allowed substitution hint:   𝑉(y)

Proof of Theorem eldm2g
StepHypRef Expression
1 eldmg 4473 . 2 (A 𝑉 → (A dom By ABy))
2 df-br 3756 . . 3 (ABy ↔ ⟨A, y B)
32exbii 1493 . 2 (y AByyA, y B)
41, 3syl6bb 185 1 (A 𝑉 → (A dom ByA, y B))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wex 1378   wcel 1390  cop 3370   class class class wbr 3755  dom cdm 4288
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-dm 4298
This theorem is referenced by:  eldm2  4476  opeldmg  4483  dmfco  5184  releldm2  5753  tfrlem9  5876
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