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Theorem fsnunres 6962
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 6455 . . . 4 (𝐹 Fn 𝑆 → (𝐹𝑆) = 𝐹)
21adantr 484 . . 3 ((𝐹 Fn 𝑆 ∧ ¬ 𝑋𝑆) → (𝐹𝑆) = 𝐹)
3 ressnop0 6927 . . . 4 𝑋𝑆 → ({⟨𝑋, 𝑌⟩} ↾ 𝑆) = ∅)
43adantl 485 . . 3 ((𝐹 Fn 𝑆 ∧ ¬ 𝑋𝑆) → ({⟨𝑋, 𝑌⟩} ↾ 𝑆) = ∅)
52, 4uneq12d 4054 . 2 ((𝐹 Fn 𝑆 ∧ ¬ 𝑋𝑆) → ((𝐹𝑆) ∪ ({⟨𝑋, 𝑌⟩} ↾ 𝑆)) = (𝐹 ∪ ∅))
6 resundir 5840 . 2 ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = ((𝐹𝑆) ∪ ({⟨𝑋, 𝑌⟩} ↾ 𝑆))
7 un0 4279 . . 3 (𝐹 ∪ ∅) = 𝐹
87eqcomi 2747 . 2 𝐹 = (𝐹 ∪ ∅)
95, 6, 83eqtr4g 2798 1 ((𝐹 Fn 𝑆 ∧ ¬ 𝑋𝑆) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1542  wcel 2114  cun 3841  c0 4211  {csn 4516  cop 4522  cres 5527   Fn wfn 6334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-xp 5531  df-rel 5532  df-dm 5535  df-res 5537  df-fun 6341  df-fn 6342
This theorem is referenced by:  pgpfaclem1  19324  islindf4  20656
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