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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 1135 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → 𝐹:(𝑆 ∖ {𝑋})⟶𝑇)
2 simp2 1136 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → 𝑋𝑆)
3 neldifsnd 4798 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → ¬ 𝑋 ∈ (𝑆 ∖ {𝑋}))
4 simp3 1137 . . 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 1374 . 2 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):((𝑆 ∖ {𝑋}) ∪ {𝑋})⟶𝑇)
7 difsnid 4815 . . . 4 (𝑋𝑆 → ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = 𝑆)
873ad2ant2 1133 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = 𝑆)
98feq2d 6723 . 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 1086   = wceq 1537  wcel 2106  cdif 3960  cun 3961  {csn 4631  cop 4637  wf 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570
This theorem is referenced by:  fsets  17203  islindf4  21876
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