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Mirrors > Home > ILE Home > Th. List > fssresd | GIF version |
Description: Restriction of a function with a subclass of its domain, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
fssresd.1 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
fssresd.2 | ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
Ref | Expression |
---|---|
fssresd | ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fssresd.1 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | fssresd.2 | . 2 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
3 | fssres 5410 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵) | |
4 | 1, 2, 3 | syl2anc 411 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊆ wss 3144 ↾ cres 4646 ⟶wf 5231 |
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-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-fun 5237 df-fn 5238 df-f 5239 |
This theorem is referenced by: cnrest 14192 cnptopresti 14195 cnptoprest 14196 psmetres2 14290 xmetres2 14336 metres2 14338 xmetresbl 14397 rescncf 14525 trilpolemlt1 15248 |
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