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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 5912 | . 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 206 ∃wex 1775 ∈ wcel 2105 Vcvv 3477 〈cop 4636 dom cdm 5688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-dm 5698 |
This theorem is referenced by: dmss 5915 opeldm 5920 dmin 5924 dmiun 5926 dmuni 5927 dm0 5933 reldm0 5940 dmrnssfld 5986 dmcoss 5987 dmcosseq 5989 dmcosseqOLD 5990 dmres 6031 iss 6054 dmsnopg 6234 relssdmrnOLD 6290 funssres 6611 dmfco 7004 fiun 7965 f1iun 7966 frrlem8 8316 frrlem10 8318 wfrlem12OLD 8358 axdc3lem2 10488 fnpr2ob 17604 gsum2d2 20006 cnlnssadj 32108 prsdm 33874 eldm3 35740 dfdm5 35753 iss2 38325 |
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