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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 2946 | . . . . 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 4849 | . . 3 ⊢ (𝐹:A⟶B ↔ (𝐹 Fn A ∧ ran 𝐹 ⊆ B)) | |
5 | df-f 4849 | . . 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 2911 ran crn 4289 Fn wfn 4840 ⟶wf 4841 |
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 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-in 2918 df-ss 2925 df-f 4849 |
This theorem is referenced by: fssd 4998 fconst6g 5028 f1ss 5040 ffoss 5101 fsn2 5280 ofco 5671 tposf2 5824 issmo2 5845 smoiso 5858 ssdomg 6194 1fv 8766 |
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