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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 5762 | . 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 208 ∃wex 1776 ∈ wcel 2110 Vcvv 3494 〈cop 4566 dom cdm 5549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-dm 5559 |
This theorem is referenced by: dmss 5765 opeldm 5770 dmin 5774 dmiun 5776 dmuni 5777 dm0 5784 reldm0 5792 dmrnssfld 5835 dmcoss 5836 dmcosseq 5838 dmres 5869 iss 5897 dmsnopg 6064 relssdmrn 6115 funssres 6392 dmfco 6751 fiun 7638 f1iun 7639 wfrlem12 7960 axdc3lem2 9867 fnpr2ob 16825 gsum2d2 19088 cnlnssadj 29851 prsdm 31152 eldm3 32992 dfdm5 33011 frrlem8 33125 frrlem10 33127 iss2 35595 |
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