| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 5909 | . 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 206 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 dom cdm 5685 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-dm 5695 |
| This theorem is referenced by: dmi 5932 dmep 5934 dmxp 5939 dmcoss 5985 dmcosseq 5987 dmcosseqOLD 5988 dminss 6173 dmsnn0 6227 dffun7 6593 dffun8 6594 fnres 6695 opabiota 6991 fndmdif 7062 dff3 7120 frxp 8151 suppvalbr 8189 reldmtpos 8259 dmtpos 8263 aceq3lem 10160 axdc2lem 10488 axdclem2 10560 fpwwe2lem11 10681 nqerf 10970 shftdm 15110 bcthlem4 25361 dchrisumlem3 27535 eulerpath 30260 fundmpss 35767 elfix 35904 fnsingle 35920 fnimage 35930 funpartlem 35943 dfrecs2 35951 dfrdg4 35952 knoppcnlem9 36502 prtlem16 38870 undmrnresiss 43617 |
| Copyright terms: Public domain | W3C validator |