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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 6192 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
2 | dmexg 7245 | . . . 4 ⊢ (𝐹 ∈ 𝐶 → dom 𝐹 ∈ V) | |
3 | eleq1 2838 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V)) | |
4 | 2, 3 | syl5ib 234 | . . 3 ⊢ (dom 𝐹 = 𝐴 → (𝐹 ∈ 𝐶 → 𝐴 ∈ V)) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∈ 𝐶 → 𝐴 ∈ V)) |
6 | 5 | impcom 394 | 1 ⊢ ((𝐹 ∈ 𝐶 ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 Vcvv 3351 dom cdm 5250 ⟶wf 6028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pr 5035 ax-un 7097 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-nul 4065 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-cnv 5258 df-dm 5260 df-rn 5261 df-fn 6035 df-f 6036 |
This theorem is referenced by: wemoiso 7301 fopwdom 8225 fowdom 8633 wdomfil 9085 fin23lem17 9363 fin23lem32 9369 fin23lem39 9375 enfin1ai 9409 fin1a2lem7 9431 lindfmm 20384 kelac1 38160 |
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