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Theorem fsnunres 7208
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 6687 . . . 4 (𝐹 Fn 𝑆 → (𝐹𝑆) = 𝐹)
21adantr 480 . . 3 ((𝐹 Fn 𝑆 ∧ ¬ 𝑋𝑆) → (𝐹𝑆) = 𝐹)
3 ressnop0 7173 . . . 4 𝑋𝑆 → ({⟨𝑋, 𝑌⟩} ↾ 𝑆) = ∅)
43adantl 481 . . 3 ((𝐹 Fn 𝑆 ∧ ¬ 𝑋𝑆) → ({⟨𝑋, 𝑌⟩} ↾ 𝑆) = ∅)
52, 4uneq12d 4169 . 2 ((𝐹 Fn 𝑆 ∧ ¬ 𝑋𝑆) → ((𝐹𝑆) ∪ ({⟨𝑋, 𝑌⟩} ↾ 𝑆)) = (𝐹 ∪ ∅))
6 resundir 6012 . 2 ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = ((𝐹𝑆) ∪ ({⟨𝑋, 𝑌⟩} ↾ 𝑆))
7 un0 4394 . . 3 (𝐹 ∪ ∅) = 𝐹
87eqcomi 2746 . 2 𝐹 = (𝐹 ∪ ∅)
95, 6, 83eqtr4g 2802 1 ((𝐹 Fn 𝑆 ∧ ¬ 𝑋𝑆) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  cun 3949  c0 4333  {csn 4626  cop 4632  cres 5687   Fn wfn 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-dm 5695  df-res 5697  df-fun 6563  df-fn 6564
This theorem is referenced by:  pgpfaclem1  20101  islindf4  21858
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