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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 6688 | . . . 4 ⊢ (𝐹 Fn 𝑆 → (𝐹 ↾ 𝑆) = 𝐹) | |
2 | 1 | adantr 480 | . . 3 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → (𝐹 ↾ 𝑆) = 𝐹) |
3 | ressnop0 7173 | . . . 4 ⊢ (¬ 𝑋 ∈ 𝑆 → ({〈𝑋, 𝑌〉} ↾ 𝑆) = ∅) | |
4 | 3 | adantl 481 | . . 3 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ({〈𝑋, 𝑌〉} ↾ 𝑆) = ∅) |
5 | 2, 4 | uneq12d 4179 | . 2 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ((𝐹 ↾ 𝑆) ∪ ({〈𝑋, 𝑌〉} ↾ 𝑆)) = (𝐹 ∪ ∅)) |
6 | resundir 6015 | . 2 ⊢ ((𝐹 ∪ {〈𝑋, 𝑌〉}) ↾ 𝑆) = ((𝐹 ↾ 𝑆) ∪ ({〈𝑋, 𝑌〉} ↾ 𝑆)) | |
7 | un0 4400 | . . 3 ⊢ (𝐹 ∪ ∅) = 𝐹 | |
8 | 7 | eqcomi 2744 | . 2 ⊢ 𝐹 = (𝐹 ∪ ∅) |
9 | 5, 6, 8 | 3eqtr4g 2800 | 1 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ((𝐹 ∪ {〈𝑋, 𝑌〉}) ↾ 𝑆) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 ∅c0 4339 {csn 4631 〈cop 4637 ↾ cres 5691 Fn wfn 6558 |
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-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-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 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-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-dm 5699 df-res 5701 df-fun 6565 df-fn 6566 |
This theorem is referenced by: pgpfaclem1 20116 islindf4 21876 |
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