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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 6387 | . . . 4 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
2 | fnressn 6922 | . . . 4 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐹 ↾ {𝐵}) = {〈𝐵, (𝐹‘𝐵)〉}) | |
3 | 1, 2 | sylanb 583 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐹 ↾ {𝐵}) = {〈𝐵, (𝐹‘𝐵)〉}) |
4 | eqimss 4025 | . . 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 4649 | . . . . 5 ⊢ ((dom 𝐹 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ dom 𝐹) | |
7 | fnresdisj 6469 | . . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → ((dom 𝐹 ∩ {𝐵}) = ∅ ↔ (𝐹 ↾ {𝐵}) = ∅)) | |
8 | 1, 7 | sylbi 219 | . . . . 5 ⊢ (Fun 𝐹 → ((dom 𝐹 ∩ {𝐵}) = ∅ ↔ (𝐹 ↾ {𝐵}) = ∅)) |
9 | 6, 8 | syl5bbr 287 | . . . 4 ⊢ (Fun 𝐹 → (¬ 𝐵 ∈ dom 𝐹 ↔ (𝐹 ↾ {𝐵}) = ∅)) |
10 | 9 | biimpa 479 | . . 3 ⊢ ((Fun 𝐹 ∧ ¬ 𝐵 ∈ dom 𝐹) → (𝐹 ↾ {𝐵}) = ∅) |
11 | 0ss 4352 | . . 3 ⊢ ∅ ⊆ {〈𝐵, (𝐹‘𝐵)〉} | |
12 | 10, 11 | eqsstrdi 4023 | . 2 ⊢ ((Fun 𝐹 ∧ ¬ 𝐵 ∈ dom 𝐹) → (𝐹 ↾ {𝐵}) ⊆ {〈𝐵, (𝐹‘𝐵)〉}) |
13 | 5, 12 | pm2.61dan 811 | 1 ⊢ (Fun 𝐹 → (𝐹 ↾ {𝐵}) ⊆ {〈𝐵, (𝐹‘𝐵)〉}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 {csn 4569 〈cop 4575 dom cdm 5557 ↾ cres 5559 Fun wfun 6351 Fn wfn 6352 ‘cfv 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 |
This theorem is referenced by: fnsnr 6929 tfrlem16 8031 fnfi 8798 fodomfi 8799 |
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