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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 6468 | . . . 4 ⊢ (𝐹 Fn 𝑆 → (𝐹 ↾ 𝑆) = 𝐹) | |
2 | 1 | adantr 483 | . . 3 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → (𝐹 ↾ 𝑆) = 𝐹) |
3 | ressnop0 6917 | . . . 4 ⊢ (¬ 𝑋 ∈ 𝑆 → ({〈𝑋, 𝑌〉} ↾ 𝑆) = ∅) | |
4 | 3 | adantl 484 | . . 3 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ({〈𝑋, 𝑌〉} ↾ 𝑆) = ∅) |
5 | 2, 4 | uneq12d 4142 | . 2 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ((𝐹 ↾ 𝑆) ∪ ({〈𝑋, 𝑌〉} ↾ 𝑆)) = (𝐹 ∪ ∅)) |
6 | resundir 5870 | . 2 ⊢ ((𝐹 ∪ {〈𝑋, 𝑌〉}) ↾ 𝑆) = ((𝐹 ↾ 𝑆) ∪ ({〈𝑋, 𝑌〉} ↾ 𝑆)) | |
7 | un0 4346 | . . 3 ⊢ (𝐹 ∪ ∅) = 𝐹 | |
8 | 7 | eqcomi 2832 | . 2 ⊢ 𝐹 = (𝐹 ∪ ∅) |
9 | 5, 6, 8 | 3eqtr4g 2883 | 1 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ((𝐹 ∪ {〈𝑋, 𝑌〉}) ↾ 𝑆) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∪ cun 3936 ∅c0 4293 {csn 4569 〈cop 4575 ↾ cres 5559 Fn wfn 6352 |
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-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 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-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-dm 5567 df-res 5569 df-fun 6359 df-fn 6360 |
This theorem is referenced by: pgpfaclem1 19205 islindf4 20984 |
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