![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > funressn | Structured version Visualization version GIF version |
Description: A function restricted to a singleton. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
funressn | ⊢ (Fun 𝐹 → (𝐹 ↾ {𝐵}) ⊆ {⟨𝐵, (𝐹‘𝐵)⟩}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funfn 6535 | . . . 4 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
2 | fnressn 7108 | . . . 4 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹‘𝐵)⟩}) | |
3 | 1, 2 | sylanb 582 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹‘𝐵)⟩}) |
4 | eqimss 4004 | . . 3 ⊢ ((𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹‘𝐵)⟩} → (𝐹 ↾ {𝐵}) ⊆ {⟨𝐵, (𝐹‘𝐵)⟩}) | |
5 | 3, 4 | syl 17 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐹 ↾ {𝐵}) ⊆ {⟨𝐵, (𝐹‘𝐵)⟩}) |
6 | disjsn 4676 | . . . . 5 ⊢ ((dom 𝐹 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ dom 𝐹) | |
7 | fnresdisj 6625 | . . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → ((dom 𝐹 ∩ {𝐵}) = ∅ ↔ (𝐹 ↾ {𝐵}) = ∅)) | |
8 | 1, 7 | sylbi 216 | . . . . 5 ⊢ (Fun 𝐹 → ((dom 𝐹 ∩ {𝐵}) = ∅ ↔ (𝐹 ↾ {𝐵}) = ∅)) |
9 | 6, 8 | bitr3id 285 | . . . 4 ⊢ (Fun 𝐹 → (¬ 𝐵 ∈ dom 𝐹 ↔ (𝐹 ↾ {𝐵}) = ∅)) |
10 | 9 | biimpa 478 | . . 3 ⊢ ((Fun 𝐹 ∧ ¬ 𝐵 ∈ dom 𝐹) → (𝐹 ↾ {𝐵}) = ∅) |
11 | 0ss 4360 | . . 3 ⊢ ∅ ⊆ {⟨𝐵, (𝐹‘𝐵)⟩} | |
12 | 10, 11 | eqsstrdi 4002 | . 2 ⊢ ((Fun 𝐹 ∧ ¬ 𝐵 ∈ dom 𝐹) → (𝐹 ↾ {𝐵}) ⊆ {⟨𝐵, (𝐹‘𝐵)⟩}) |
13 | 5, 12 | pm2.61dan 812 | 1 ⊢ (Fun 𝐹 → (𝐹 ↾ {𝐵}) ⊆ {⟨𝐵, (𝐹‘𝐵)⟩}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∩ cin 3913 ⊆ wss 3914 ∅c0 4286 {csn 4590 ⟨cop 4596 dom cdm 5637 ↾ cres 5639 Fun wfun 6494 Fn wfn 6495 ‘cfv 6500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 |
This theorem is referenced by: fnsnr 7115 tfrlem16 8343 fnfi 9131 fodomfi 9275 |
Copyright terms: Public domain | W3C validator |