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Theorem fsnunf2 7206
Description: Adjoining a point to a punctured function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Assertion
Ref Expression
fsnunf2 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):𝑆𝑇)

Proof of Theorem fsnunf2
StepHypRef Expression
1 simp1 1137 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → 𝐹:(𝑆 ∖ {𝑋})⟶𝑇)
2 simp2 1138 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → 𝑋𝑆)
3 neldifsnd 4793 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → ¬ 𝑋 ∈ (𝑆 ∖ {𝑋}))
4 simp3 1139 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → 𝑌𝑇)
5 fsnunf 7205 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇 ∧ (𝑋𝑆 ∧ ¬ 𝑋 ∈ (𝑆 ∖ {𝑋})) ∧ 𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):((𝑆 ∖ {𝑋}) ∪ {𝑋})⟶𝑇)
61, 2, 3, 4, 5syl121anc 1377 . 2 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):((𝑆 ∖ {𝑋}) ∪ {𝑋})⟶𝑇)
7 difsnid 4810 . . . 4 (𝑋𝑆 → ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = 𝑆)
873ad2ant2 1135 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = 𝑆)
98feq2d 6722 . 2 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}):((𝑆 ∖ {𝑋}) ∪ {𝑋})⟶𝑇 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}):𝑆𝑇))
106, 9mpbid 232 1 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):𝑆𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  w3a 1087   = wceq 1540  wcel 2108  cdif 3948  cun 3949  {csn 4626  cop 4632  wf 6557
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-10 2141  ax-12 2177  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-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  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-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568
This theorem is referenced by:  fsets  17206  islindf4  21858
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