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Theorem fsnunres 5307
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 ¬ 𝑋 𝑆) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = 𝐹)

Proof of Theorem fsnunres
StepHypRef Expression
1 fnresdm 4951 . . . 4 (𝐹 Fn 𝑆 → (𝐹𝑆) = 𝐹)
21adantr 261 . . 3 ((𝐹 Fn 𝑆 ¬ 𝑋 𝑆) → (𝐹𝑆) = 𝐹)
3 ressnop0 5287 . . . 4 𝑋 𝑆 → ({⟨𝑋, 𝑌⟩} ↾ 𝑆) = ∅)
43adantl 262 . . 3 ((𝐹 Fn 𝑆 ¬ 𝑋 𝑆) → ({⟨𝑋, 𝑌⟩} ↾ 𝑆) = ∅)
52, 4uneq12d 3092 . 2 ((𝐹 Fn 𝑆 ¬ 𝑋 𝑆) → ((𝐹𝑆) ∪ ({⟨𝑋, 𝑌⟩} ↾ 𝑆)) = (𝐹 ∪ ∅))
6 resundir 4569 . 2 ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = ((𝐹𝑆) ∪ ({⟨𝑋, 𝑌⟩} ↾ 𝑆))
7 un0 3245 . . 3 (𝐹 ∪ ∅) = 𝐹
87eqcomi 2041 . 2 𝐹 = (𝐹 ∪ ∅)
95, 6, 83eqtr4g 2094 1 ((𝐹 Fn 𝑆 ¬ 𝑋 𝑆) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = 𝐹)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = wceq 1242   wcel 1390  cun 2909  c0 3218  {csn 3367  cop 3370  cres 4290   Fn wfn 4840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-dm 4298  df-res 4300  df-fun 4847  df-fn 4848
This theorem is referenced by:  tfrlemisucaccv  5880
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