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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 5273 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
2 | dmexg 4798 | . . . 4 ⊢ (𝐹 ∈ 𝐶 → dom 𝐹 ∈ V) | |
3 | eleq1 2200 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V)) | |
4 | 2, 3 | syl5ib 153 | . . 3 ⊢ (dom 𝐹 = 𝐴 → (𝐹 ∈ 𝐶 → 𝐴 ∈ V)) |
5 | 1, 4 | syl 14 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∈ 𝐶 → 𝐴 ∈ V)) |
6 | 5 | impcom 124 | 1 ⊢ ((𝐹 ∈ 𝐶 ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 Vcvv 2681 dom cdm 4534 ⟶wf 5114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-cnv 4542 df-dm 4544 df-rn 4545 df-fn 5121 df-f 5122 |
This theorem is referenced by: fopwdom 6723 |
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