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Mirrors > Home > ILE Home > Th. List > eldm2 | GIF version |
Description: Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 1-Aug-1994.) |
Ref | Expression |
---|---|
eldm.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
eldm2 | ⊢ (𝐴 ∈ dom 𝐵 ↔ ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldm.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | eldm2g 4531 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐵)) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (𝐴 ∈ dom 𝐵 ↔ ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∃wex 1381 ∈ wcel 1393 Vcvv 2557 〈cop 3378 dom cdm 4345 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-dm 4355 |
This theorem is referenced by: dmss 4534 opeldm 4538 dmin 4543 dmiun 4544 dmuni 4545 dm0 4549 reldm0 4553 dmrnssfld 4595 dmcoss 4601 dmcosseq 4603 dmres 4632 iss 4654 dmxpss 4753 dmsnopg 4792 relssdmrn 4841 funssres 4942 fun11iun 5147 tfrlemibxssdm 5941 |
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