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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 3204 | . . . . 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 5289 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
| 5 | df-f 5289 | . . 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 3170 ran crn 4689 Fn wfn 5280 ⟶wf 5281 |
| 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 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-11 1530 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-ext 2188 |
| This theorem depends on definitions: df-bi 117 df-nf 1485 df-sb 1787 df-clab 2193 df-cleq 2199 df-clel 2202 df-in 3176 df-ss 3183 df-f 5289 |
| This theorem is referenced by: fssd 5453 fconst6g 5491 f1ss 5504 ffoss 5571 fsn2 5772 ofco 6195 tposf2 6372 issmo2 6393 smoiso 6406 mapsn 6795 ssdomg 6888 omp1eomlem 7217 1fv 10291 fxnn0nninf 10616 abscn2 11711 recn2 11713 imcn2 11714 climabs 11716 climre 11718 climim 11719 fsumre 11868 fsumim 11869 resmhm2 13405 prdsgrpd 13526 prdsinvgd 13527 ismet2 14911 dvfre 15267 dvrecap 15270 elplyr 15297 lgsfcl 15570 |
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