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Theorem eldmg 4473
 Description: Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
eldmg (A 𝑉 → (A dom By ABy))
Distinct variable groups:   y,A   y,B
Allowed substitution hint:   𝑉(y)

Proof of Theorem eldmg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 breq1 3758 . . 3 (x = A → (xByABy))
21exbidv 1703 . 2 (x = A → (y xByy ABy))
3 df-dm 4298 . 2 dom B = {xy xBy}
42, 3elab2g 2683 1 (A 𝑉 → (A dom By ABy))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390   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-bndl 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:  eldm2g  4474  eldm  4475  breldmg  4484  releldmb  4514  funeu  4869  fneu  4946  ndmfvg  5147  erref  6062  ecdmn0m  6084
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