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Theorem fsnunres 6359
 Description: Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Assertion
Ref Expression
fsnunres ((𝐹 Fn 𝑆 ∧ ¬ 𝑋𝑆) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = 𝐹)

Proof of Theorem fsnunres
StepHypRef Expression
1 fnresdm 5914 . . . 4 (𝐹 Fn 𝑆 → (𝐹𝑆) = 𝐹)
21adantr 480 . . 3 ((𝐹 Fn 𝑆 ∧ ¬ 𝑋𝑆) → (𝐹𝑆) = 𝐹)
3 ressnop0 6325 . . . 4 𝑋𝑆 → ({⟨𝑋, 𝑌⟩} ↾ 𝑆) = ∅)
43adantl 481 . . 3 ((𝐹 Fn 𝑆 ∧ ¬ 𝑋𝑆) → ({⟨𝑋, 𝑌⟩} ↾ 𝑆) = ∅)
52, 4uneq12d 3730 . 2 ((𝐹 Fn 𝑆 ∧ ¬ 𝑋𝑆) → ((𝐹𝑆) ∪ ({⟨𝑋, 𝑌⟩} ↾ 𝑆)) = (𝐹 ∪ ∅))
6 resundir 5331 . 2 ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = ((𝐹𝑆) ∪ ({⟨𝑋, 𝑌⟩} ↾ 𝑆))
7 un0 3919 . . 3 (𝐹 ∪ ∅) = 𝐹
87eqcomi 2619 . 2 𝐹 = (𝐹 ∪ ∅)
95, 6, 83eqtr4g 2669 1 ((𝐹 Fn 𝑆 ∧ ¬ 𝑋𝑆) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = 𝐹)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∪ cun 3538  ∅c0 3874  {csn 4125  ⟨cop 4131   ↾ cres 5040   Fn wfn 5799 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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  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-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-dm 5048  df-res 5050  df-fun 5806  df-fn 5807 This theorem is referenced by:  pgpfaclem1  18303  islindf4  19996
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