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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 5833 | . . . 4 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
2 | fnressn 6330 | . . . 4 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐹 ↾ {𝐵}) = {〈𝐵, (𝐹‘𝐵)〉}) | |
3 | 1, 2 | sylanb 488 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐹 ↾ {𝐵}) = {〈𝐵, (𝐹‘𝐵)〉}) |
4 | eqimss 3620 | . . 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 4192 | . . . . 5 ⊢ ((dom 𝐹 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ dom 𝐹) | |
7 | fnresdisj 5915 | . . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → ((dom 𝐹 ∩ {𝐵}) = ∅ ↔ (𝐹 ↾ {𝐵}) = ∅)) | |
8 | 1, 7 | sylbi 206 | . . . . 5 ⊢ (Fun 𝐹 → ((dom 𝐹 ∩ {𝐵}) = ∅ ↔ (𝐹 ↾ {𝐵}) = ∅)) |
9 | 6, 8 | syl5bbr 273 | . . . 4 ⊢ (Fun 𝐹 → (¬ 𝐵 ∈ dom 𝐹 ↔ (𝐹 ↾ {𝐵}) = ∅)) |
10 | 9 | biimpa 500 | . . 3 ⊢ ((Fun 𝐹 ∧ ¬ 𝐵 ∈ dom 𝐹) → (𝐹 ↾ {𝐵}) = ∅) |
11 | 0ss 3924 | . . 3 ⊢ ∅ ⊆ {〈𝐵, (𝐹‘𝐵)〉} | |
12 | 10, 11 | syl6eqss 3618 | . 2 ⊢ ((Fun 𝐹 ∧ ¬ 𝐵 ∈ dom 𝐹) → (𝐹 ↾ {𝐵}) ⊆ {〈𝐵, (𝐹‘𝐵)〉}) |
13 | 5, 12 | pm2.61dan 828 | 1 ⊢ (Fun 𝐹 → (𝐹 ↾ {𝐵}) ⊆ {〈𝐵, (𝐹‘𝐵)〉}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 〈cop 4131 dom cdm 5038 ↾ cres 5040 Fun wfun 5798 Fn wfn 5799 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 |
This theorem is referenced by: fnsnb 6337 tfrlem16 7376 fnfi 8123 fodomfi 8124 bnj142OLD 30048 |
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