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Theorem fsnunres 5718
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 5325 . . . 4 (𝐹 Fn 𝑆 → (𝐹𝑆) = 𝐹)
21adantr 276 . . 3 ((𝐹 Fn 𝑆 ∧ ¬ 𝑋𝑆) → (𝐹𝑆) = 𝐹)
3 ressnop0 5697 . . . 4 𝑋𝑆 → ({⟨𝑋, 𝑌⟩} ↾ 𝑆) = ∅)
43adantl 277 . . 3 ((𝐹 Fn 𝑆 ∧ ¬ 𝑋𝑆) → ({⟨𝑋, 𝑌⟩} ↾ 𝑆) = ∅)
52, 4uneq12d 3290 . 2 ((𝐹 Fn 𝑆 ∧ ¬ 𝑋𝑆) → ((𝐹𝑆) ∪ ({⟨𝑋, 𝑌⟩} ↾ 𝑆)) = (𝐹 ∪ ∅))
6 resundir 4921 . 2 ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = ((𝐹𝑆) ∪ ({⟨𝑋, 𝑌⟩} ↾ 𝑆))
7 un0 3456 . . 3 (𝐹 ∪ ∅) = 𝐹
87eqcomi 2181 . 2 𝐹 = (𝐹 ∪ ∅)
95, 6, 83eqtr4g 2235 1 ((𝐹 Fn 𝑆 ∧ ¬ 𝑋𝑆) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = 𝐹)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104   = wceq 1353  wcel 2148  cun 3127  c0 3422  {csn 3592  cop 3595  cres 4628   Fn wfn 5211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-dm 4636  df-res 4638  df-fun 5218  df-fn 5219
This theorem is referenced by:  tfrlemisucaccv  6325  tfr1onlemsucaccv  6341  tfrcllemsucaccv  6354
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