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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 3006 | . . . . 5 ⊢ (ran 𝐹 ⊆ 𝐵 → (𝐵 ⊆ 𝐶 → ran 𝐹 ⊆ 𝐶)) | |
2 | 1 | com12 30 | . . . 4 ⊢ (𝐵 ⊆ 𝐶 → (ran 𝐹 ⊆ 𝐵 → ran 𝐹 ⊆ 𝐶)) |
3 | 2 | anim2d 330 | . . 3 ⊢ (𝐵 ⊆ 𝐶 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))) |
4 | df-f 4926 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
5 | df-f 4926 | . . 3 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶)) | |
6 | 3, 4, 5 | 3imtr4g 203 | . 2 ⊢ (𝐵 ⊆ 𝐶 → (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶𝐶)) |
7 | 6 | impcom 123 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ⊆ wss 2973 ran crn 4364 Fn wfn 4917 ⟶wf 4918 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-11 1437 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-in 2979 df-ss 2986 df-f 4926 |
This theorem is referenced by: fssd 5075 fconst6g 5105 f1ss 5117 ffoss 5178 fsn2 5358 ofco 5749 tposf2 5906 issmo2 5927 smoiso 5940 ssdomg 6281 1fv 9149 abscn2 10153 recn2 10155 imcn2 10156 climabs 10158 climre 10160 climim 10161 |
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