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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 3054 | . . . . 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 333 | . . 3 ⊢ (𝐵 ⊆ 𝐶 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))) |
4 | df-f 5063 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
5 | df-f 5063 | . . 3 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶)) | |
6 | 3, 4, 5 | 3imtr4g 204 | . 2 ⊢ (𝐵 ⊆ 𝐶 → (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶𝐶)) |
7 | 6 | impcom 124 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ⊆ wss 3021 ran crn 4478 Fn wfn 5054 ⟶wf 5055 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-11 1452 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-in 3027 df-ss 3034 df-f 5063 |
This theorem is referenced by: fssd 5221 fconst6g 5257 f1ss 5270 ffoss 5333 fsn2 5526 ofco 5931 tposf2 6095 issmo2 6116 smoiso 6129 mapsn 6514 ssdomg 6602 omp1eomlem 6894 1fv 9757 fxnn0nninf 10052 abscn2 10923 recn2 10925 imcn2 10926 climabs 10928 climre 10930 climim 10931 fsumre 11080 fsumim 11081 ismet2 12282 |
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