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Mirrors > Home > ILE Home > Th. List > dmfex | 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 5050 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
2 | dmexg 4596 | . . . 4 ⊢ (𝐹 ∈ 𝐶 → dom 𝐹 ∈ V) | |
3 | eleq1 2100 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V)) | |
4 | 2, 3 | syl5ib 143 | . . 3 ⊢ (dom 𝐹 = 𝐴 → (𝐹 ∈ 𝐶 → 𝐴 ∈ V)) |
5 | 1, 4 | syl 14 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∈ 𝐶 → 𝐴 ∈ V)) |
6 | 5 | impcom 116 | 1 ⊢ ((𝐹 ∈ 𝐶 ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 Vcvv 2557 dom cdm 4345 ⟶wf 4898 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-cnv 4353 df-dm 4355 df-rn 4356 df-fn 4905 df-f 4906 |
This theorem is referenced by: fopwdom 6310 |
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