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Mirrors > Home > MPE Home > Th. List > dmfex | Structured version Visualization version GIF version |
Description: If a mapping is a set, its domain is a set. (Contributed by NM, 27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
dmfex | ⊢ ((𝐹 ∈ 𝐶 ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fdm 6717 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
2 | dmexg 7888 | . . . 4 ⊢ (𝐹 ∈ 𝐶 → dom 𝐹 ∈ V) | |
3 | eleq1 2813 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V)) | |
4 | 2, 3 | imbitrid 243 | . . 3 ⊢ (dom 𝐹 = 𝐴 → (𝐹 ∈ 𝐶 → 𝐴 ∈ V)) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∈ 𝐶 → 𝐴 ∈ V)) |
6 | 5 | impcom 407 | 1 ⊢ ((𝐹 ∈ 𝐶 ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3466 dom cdm 5667 ⟶wf 6530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-cnv 5675 df-dm 5677 df-rn 5678 df-fn 6537 df-f 6538 |
This theorem is referenced by: wemoiso 7954 mapfset 8841 mapfoss 8843 fopwdom 9077 fowdom 9563 wdomfil 10053 fin23lem17 10330 fin23lem32 10336 fin23lem39 10342 enfin1ai 10376 fin1a2lem7 10398 symgbasmap 19292 lindfmm 21711 kelac1 42357 |
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