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Mirrors > Home > MPE Home > Th. List > fsnunres | Structured version Visualization version GIF version |
Description: Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
Ref | Expression |
---|---|
fsnunres | ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ((𝐹 ∪ {〈𝑋, 𝑌〉}) ↾ 𝑆) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnresdm 6438 | . . . 4 ⊢ (𝐹 Fn 𝑆 → (𝐹 ↾ 𝑆) = 𝐹) | |
2 | 1 | adantr 484 | . . 3 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → (𝐹 ↾ 𝑆) = 𝐹) |
3 | ressnop0 6892 | . . . 4 ⊢ (¬ 𝑋 ∈ 𝑆 → ({〈𝑋, 𝑌〉} ↾ 𝑆) = ∅) | |
4 | 3 | adantl 485 | . . 3 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ({〈𝑋, 𝑌〉} ↾ 𝑆) = ∅) |
5 | 2, 4 | uneq12d 4091 | . 2 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ((𝐹 ↾ 𝑆) ∪ ({〈𝑋, 𝑌〉} ↾ 𝑆)) = (𝐹 ∪ ∅)) |
6 | resundir 5833 | . 2 ⊢ ((𝐹 ∪ {〈𝑋, 𝑌〉}) ↾ 𝑆) = ((𝐹 ↾ 𝑆) ∪ ({〈𝑋, 𝑌〉} ↾ 𝑆)) | |
7 | un0 4298 | . . 3 ⊢ (𝐹 ∪ ∅) = 𝐹 | |
8 | 7 | eqcomi 2807 | . 2 ⊢ 𝐹 = (𝐹 ∪ ∅) |
9 | 5, 6, 8 | 3eqtr4g 2858 | 1 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ((𝐹 ∪ {〈𝑋, 𝑌〉}) ↾ 𝑆) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∪ cun 3879 ∅c0 4243 {csn 4525 〈cop 4531 ↾ cres 5521 Fn wfn 6319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-dm 5529 df-res 5531 df-fun 6326 df-fn 6327 |
This theorem is referenced by: pgpfaclem1 19196 islindf4 20527 |
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