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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 5895 | . 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 205 ∃wex 1774 ∈ wcel 2099 Vcvv 3470 class class class wbr 5142 dom cdm 5672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-dm 5682 |
This theorem is referenced by: dmi 5918 dmep 5920 dmcoss 5968 dmcosseq 5970 dminss 6151 dmsnn0 6205 dffun7 6574 dffun8 6575 fnres 6676 opabiota 6975 fndmdif 7045 dff3 7104 frxp 8125 suppvalbr 8163 reldmtpos 8233 dmtpos 8237 aceq3lem 10137 axdc2lem 10465 axdclem2 10537 fpwwe2lem11 10658 nqerf 10947 shftdm 15044 bcthlem4 25248 dchrisumlem3 27417 eulerpath 30044 fundmpss 35356 elfix 35493 fnsingle 35509 fnimage 35519 funpartlem 35532 dfrecs2 35540 dfrdg4 35541 knoppcnlem9 35970 prtlem16 38335 undmrnresiss 43028 |
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