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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 6598 | . . . 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 4054 | . . 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 4716 | . . . . 5 ⊢ ((dom 𝐹 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ dom 𝐹) | |
7 | fnresdisj 6689 | . . . . . 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 4406 | . . 3 ⊢ ∅ ⊆ {〈𝐵, (𝐹‘𝐵)〉} | |
12 | 10, 11 | eqsstrdi 4050 | . 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 1537 ∈ wcel 2106 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 {csn 4631 〈cop 4637 dom cdm 5689 ↾ cres 5691 Fun wfun 6557 Fn wfn 6558 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 |
This theorem is referenced by: fnsnr 7185 tfrlem16 8432 fnfi 9216 fodomfi 9348 fodomfiOLD 9368 |
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