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Mirrors > Home > ILE Home > Th. List > eldm | GIF version |
Description: Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.) |
Ref | Expression |
---|---|
eldm.1 | ⊢ A ∈ V |
Ref | Expression |
---|---|
eldm | ⊢ (A ∈ dom B ↔ ∃y ABy) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldm.1 | . 2 ⊢ A ∈ V | |
2 | eldmg 4473 | . 2 ⊢ (A ∈ V → (A ∈ dom B ↔ ∃y ABy)) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (A ∈ dom B ↔ ∃y ABy) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∃wex 1378 ∈ wcel 1390 Vcvv 2551 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: dmi 4493 dmcoss 4544 dmcosseq 4546 dminss 4681 dmsnm 4729 dffun7 4871 dffun8 4872 fnres 4958 fndmdif 5215 reldmtpos 5809 dmtpos 5812 tfrexlem 5889 |
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