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Mirrors > Home > MPE Home > Th. List > fssres2 | Structured version Visualization version GIF version |
Description: Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.) |
Ref | Expression |
---|---|
fssres2 | ⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fssres 6070 | . 2 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → ((𝐹 ↾ 𝐴) ↾ 𝐶):𝐶⟶𝐵) | |
2 | resabs1 5427 | . . . 4 ⊢ (𝐶 ⊆ 𝐴 → ((𝐹 ↾ 𝐴) ↾ 𝐶) = (𝐹 ↾ 𝐶)) | |
3 | 2 | feq1d 6030 | . . 3 ⊢ (𝐶 ⊆ 𝐴 → (((𝐹 ↾ 𝐴) ↾ 𝐶):𝐶⟶𝐵 ↔ (𝐹 ↾ 𝐶):𝐶⟶𝐵)) |
4 | 3 | adantl 482 | . 2 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (((𝐹 ↾ 𝐴) ↾ 𝐶):𝐶⟶𝐵 ↔ (𝐹 ↾ 𝐶):𝐶⟶𝐵)) |
5 | 1, 4 | mpbid 222 | 1 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ⊆ wss 3574 ↾ cres 5116 ⟶wf 5884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-fun 5890 df-fn 5891 df-f 5892 |
This theorem is referenced by: efcvx 24203 filnetlem4 32376 |
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