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Mirrors > Home > ILE Home > Th. List > eldm2g | GIF version |
Description: Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
eldm2g | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldmg 4530 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)) | |
2 | df-br 3765 | . . 3 ⊢ (𝐴𝐵𝑦 ↔ 〈𝐴, 𝑦〉 ∈ 𝐵) | |
3 | 2 | exbii 1496 | . 2 ⊢ (∃𝑦 𝐴𝐵𝑦 ↔ ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐵) |
4 | 1, 3 | syl6bb 185 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∃wex 1381 ∈ wcel 1393 〈cop 3378 class class class wbr 3764 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: eldm2 4533 opeldmg 4540 dmfco 5241 releldm2 5811 tfrlem9 5935 climcau 9866 |
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