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Mirrors > Home > MPE Home > Th. List > eldm | Structured version Visualization version GIF version |
Description: Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.) |
Ref | Expression |
---|---|
eldm.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
eldm | ⊢ (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldm.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | eldmg 5241 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 dom cdm 5038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-dm 5048 |
This theorem is referenced by: dmi 5261 dmcoss 5306 dmcosseq 5308 dminss 5466 dmsnn0 5518 dffun7 5830 dffun8 5831 fnres 5921 opabiota 6171 fndmdif 6229 dff3 6280 frxp 7174 suppvalbr 7186 reldmtpos 7247 dmtpos 7251 aceq3lem 8826 axdc2lem 9153 axdclem2 9225 fpwwe2lem12 9342 nqerf 9631 shftdm 13659 xpsfrnel2 16048 bcthlem4 22932 dchrisumlem3 24980 eupath 26508 fundmpss 30910 elfix 31180 fnsingle 31196 fnimage 31206 funpartlem 31219 dfrecs2 31227 dfrdg4 31228 knoppcnlem9 31661 prtlem16 33172 undmrnresiss 36929 eulerpath 41409 |
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