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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 3162 | . . . . 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 337 | . . 3 ⊢ (𝐵 ⊆ 𝐶 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))) |
4 | df-f 5220 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
5 | df-f 5220 | . . 3 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶)) | |
6 | 3, 4, 5 | 3imtr4g 205 | . 2 ⊢ (𝐵 ⊆ 𝐶 → (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶𝐶)) |
7 | 6 | impcom 125 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ⊆ wss 3129 ran crn 4627 Fn wfn 5211 ⟶wf 5212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-in 3135 df-ss 3142 df-f 5220 |
This theorem is referenced by: fssd 5378 fconst6g 5414 f1ss 5427 ffoss 5493 fsn2 5690 ofco 6100 tposf2 6268 issmo2 6289 smoiso 6302 mapsn 6689 ssdomg 6777 omp1eomlem 7092 1fv 10138 fxnn0nninf 10437 abscn2 11322 recn2 11324 imcn2 11325 climabs 11327 climre 11329 climim 11330 fsumre 11479 fsumim 11480 ismet2 13824 dvfre 14144 dvrecap 14147 lgsfcl 14379 |
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