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Theorem fsnunf2 6618
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 1131 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → 𝐹:(𝑆 ∖ {𝑋})⟶𝑇)
2 simp2 1132 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → 𝑋𝑆)
3 neldifsnd 4469 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → ¬ 𝑋 ∈ (𝑆 ∖ {𝑋}))
4 simp3 1133 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → 𝑌𝑇)
5 fsnunf 6617 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇 ∧ (𝑋𝑆 ∧ ¬ 𝑋 ∈ (𝑆 ∖ {𝑋})) ∧ 𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):((𝑆 ∖ {𝑋}) ∪ {𝑋})⟶𝑇)
61, 2, 3, 4, 5syl121anc 1482 . 2 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):((𝑆 ∖ {𝑋}) ∪ {𝑋})⟶𝑇)
7 difsnid 4487 . . . 4 (𝑋𝑆 → ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = 𝑆)
873ad2ant2 1129 . . 3 ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = 𝑆)
98feq2d 6193 . 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 1072   = wceq 1632  wcel 2140  cdif 3713  cun 3714  {csn 4322  cop 4328  wf 6046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057
This theorem is referenced by:  fsets  16114  islindf4  20400
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