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Theorem fsnunf2 6493
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 1081 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → 𝐹:(𝑆 ∖ {𝑋})⟶𝑇)
2 simp2 1082 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → 𝑋𝑆)
3 neldifsnd 4355 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → ¬ 𝑋 ∈ (𝑆 ∖ {𝑋}))
4 simp3 1083 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → 𝑌𝑇)
5 fsnunf 6492 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇 ∧ (𝑋𝑆 ∧ ¬ 𝑋 ∈ (𝑆 ∖ {𝑋})) ∧ 𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):((𝑆 ∖ {𝑋}) ∪ {𝑋})⟶𝑇)
61, 2, 3, 4, 5syl121anc 1371 . 2 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):((𝑆 ∖ {𝑋}) ∪ {𝑋})⟶𝑇)
7 difsnid 4373 . . . 4 (𝑋𝑆 → ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = 𝑆)
873ad2ant2 1103 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = 𝑆)
98feq2d 6069 . 2 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}):((𝑆 ∖ {𝑋}) ∪ {𝑋})⟶𝑇 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}):𝑆𝑇))
106, 9mpbid 222 1 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):𝑆𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  w3a 1054   = wceq 1523  wcel 2030  cdif 3604  cun 3605  {csn 4210  cop 4216  wf 5922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933
This theorem is referenced by:  fsets  15938  islindf4  20225
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