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Theorem fsnunf2 7185
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 1134 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → 𝐹:(𝑆 ∖ {𝑋})⟶𝑇)
2 simp2 1135 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → 𝑋𝑆)
3 neldifsnd 4795 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → ¬ 𝑋 ∈ (𝑆 ∖ {𝑋}))
4 simp3 1136 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → 𝑌𝑇)
5 fsnunf 7184 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇 ∧ (𝑋𝑆 ∧ ¬ 𝑋 ∈ (𝑆 ∖ {𝑋})) ∧ 𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):((𝑆 ∖ {𝑋}) ∪ {𝑋})⟶𝑇)
61, 2, 3, 4, 5syl121anc 1373 . 2 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):((𝑆 ∖ {𝑋}) ∪ {𝑋})⟶𝑇)
7 difsnid 4812 . . . 4 (𝑋𝑆 → ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = 𝑆)
873ad2ant2 1132 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = 𝑆)
98feq2d 6702 . 2 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}):((𝑆 ∖ {𝑋}) ∪ {𝑋})⟶𝑇 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}):𝑆𝑇))
106, 9mpbid 231 1 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):𝑆𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  w3a 1085   = wceq 1539  wcel 2104  cdif 3944  cun 3945  {csn 4627  cop 4633  wf 6538
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549
This theorem is referenced by:  fsets  17106  islindf4  21612
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