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Mirrors > Home > MPE Home > Th. List > eldm2 | Structured version Visualization version 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 5732 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∈ dom 𝐵 ↔ ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∃wex 1781 ∈ wcel 2111 Vcvv 3441 〈cop 4531 dom cdm 5519 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-un 3886 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-dm 5529 |
This theorem is referenced by: dmss 5735 opeldm 5740 dmin 5744 dmiun 5746 dmuni 5747 dm0 5754 reldm0 5762 dmrnssfld 5806 dmcoss 5807 dmcosseq 5809 dmres 5840 iss 5870 dmsnopg 6037 relssdmrn 6088 funssres 6368 dmfco 6734 fiun 7626 f1iun 7627 wfrlem12 7949 axdc3lem2 9862 fnpr2ob 16823 gsum2d2 19087 cnlnssadj 29863 prsdm 31267 eldm3 33110 dfdm5 33129 frrlem8 33243 frrlem10 33245 iss2 35761 |
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