|   | 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 6596 | . . . 4 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
| 2 | fnressn 7178 | . . . 4 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐹 ↾ {𝐵}) = {〈𝐵, (𝐹‘𝐵)〉}) | |
| 3 | 1, 2 | sylanb 581 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐹 ↾ {𝐵}) = {〈𝐵, (𝐹‘𝐵)〉}) | 
| 4 | eqimss 4042 | . . 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 4711 | . . . . 5 ⊢ ((dom 𝐹 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ dom 𝐹) | |
| 7 | fnresdisj 6688 | . . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → ((dom 𝐹 ∩ {𝐵}) = ∅ ↔ (𝐹 ↾ {𝐵}) = ∅)) | |
| 8 | 1, 7 | sylbi 217 | . . . . 5 ⊢ (Fun 𝐹 → ((dom 𝐹 ∩ {𝐵}) = ∅ ↔ (𝐹 ↾ {𝐵}) = ∅)) | 
| 9 | 6, 8 | bitr3id 285 | . . . 4 ⊢ (Fun 𝐹 → (¬ 𝐵 ∈ dom 𝐹 ↔ (𝐹 ↾ {𝐵}) = ∅)) | 
| 10 | 9 | biimpa 476 | . . 3 ⊢ ((Fun 𝐹 ∧ ¬ 𝐵 ∈ dom 𝐹) → (𝐹 ↾ {𝐵}) = ∅) | 
| 11 | 0ss 4400 | . . 3 ⊢ ∅ ⊆ {〈𝐵, (𝐹‘𝐵)〉} | |
| 12 | 10, 11 | eqsstrdi 4028 | . 2 ⊢ ((Fun 𝐹 ∧ ¬ 𝐵 ∈ dom 𝐹) → (𝐹 ↾ {𝐵}) ⊆ {〈𝐵, (𝐹‘𝐵)〉}) | 
| 13 | 5, 12 | pm2.61dan 813 | 1 ⊢ (Fun 𝐹 → (𝐹 ↾ {𝐵}) ⊆ {〈𝐵, (𝐹‘𝐵)〉}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {csn 4626 〈cop 4632 dom cdm 5685 ↾ cres 5687 Fun wfun 6555 Fn wfn 6556 ‘cfv 6561 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 | 
| This theorem is referenced by: fnsnr 7185 tfrlem16 8433 fnfi 9218 fodomfi 9350 fodomfiOLD 9370 | 
| Copyright terms: Public domain | W3C validator |