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Mirrors > Home > ILE Home > Th. List > fss | GIF version |
Description: Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
fss | ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sstr2 2952 | . . . . 5 ⊢ (ran 𝐹 ⊆ 𝐵 → (𝐵 ⊆ 𝐶 → ran 𝐹 ⊆ 𝐶)) | |
2 | 1 | com12 27 | . . . 4 ⊢ (𝐵 ⊆ 𝐶 → (ran 𝐹 ⊆ 𝐵 → ran 𝐹 ⊆ 𝐶)) |
3 | 2 | anim2d 320 | . . 3 ⊢ (𝐵 ⊆ 𝐶 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))) |
4 | df-f 4906 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
5 | df-f 4906 | . . 3 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶)) | |
6 | 3, 4, 5 | 3imtr4g 194 | . 2 ⊢ (𝐵 ⊆ 𝐶 → (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶𝐶)) |
7 | 6 | impcom 116 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ⊆ wss 2917 ran crn 4346 Fn wfn 4897 ⟶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-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-in 2924 df-ss 2931 df-f 4906 |
This theorem is referenced by: fssd 5055 fconst6g 5085 f1ss 5097 ffoss 5158 fsn2 5337 ofco 5729 tposf2 5883 issmo2 5904 smoiso 5917 ssdomg 6258 1fv 8996 abscn2 9835 recn2 9837 imcn2 9838 climabs 9840 climre 9842 climim 9843 |
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