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Theorem fsnunf 5305
Description: Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Assertion
Ref Expression
fsnunf ((𝐹:𝑆𝑇 (𝑋 𝑉 ¬ 𝑋 𝑆) 𝑌 𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶𝑇)

Proof of Theorem fsnunf
StepHypRef Expression
1 simp1 903 . . 3 ((𝐹:𝑆𝑇 (𝑋 𝑉 ¬ 𝑋 𝑆) 𝑌 𝑇) → 𝐹:𝑆𝑇)
2 simp2l 929 . . . . 5 ((𝐹:𝑆𝑇 (𝑋 𝑉 ¬ 𝑋 𝑆) 𝑌 𝑇) → 𝑋 𝑉)
3 simp3 905 . . . . 5 ((𝐹:𝑆𝑇 (𝑋 𝑉 ¬ 𝑋 𝑆) 𝑌 𝑇) → 𝑌 𝑇)
4 f1osng 5110 . . . . 5 ((𝑋 𝑉 𝑌 𝑇) → {⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌})
52, 3, 4syl2anc 391 . . . 4 ((𝐹:𝑆𝑇 (𝑋 𝑉 ¬ 𝑋 𝑆) 𝑌 𝑇) → {⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌})
6 f1of 5069 . . . 4 ({⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌} → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
75, 6syl 14 . . 3 ((𝐹:𝑆𝑇 (𝑋 𝑉 ¬ 𝑋 𝑆) 𝑌 𝑇) → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
8 simp2r 930 . . . 4 ((𝐹:𝑆𝑇 (𝑋 𝑉 ¬ 𝑋 𝑆) 𝑌 𝑇) → ¬ 𝑋 𝑆)
9 disjsn 3423 . . . 4 ((𝑆 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 𝑆)
108, 9sylibr 137 . . 3 ((𝐹:𝑆𝑇 (𝑋 𝑉 ¬ 𝑋 𝑆) 𝑌 𝑇) → (𝑆 ∩ {𝑋}) = ∅)
11 fun 5006 . . 3 (((𝐹:𝑆𝑇 {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌}) (𝑆 ∩ {𝑋}) = ∅) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌}))
121, 7, 10, 11syl21anc 1133 . 2 ((𝐹:𝑆𝑇 (𝑋 𝑉 ¬ 𝑋 𝑆) 𝑌 𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌}))
13 snssi 3499 . . . . 5 (𝑌 𝑇 → {𝑌} ⊆ 𝑇)
14133ad2ant3 926 . . . 4 ((𝐹:𝑆𝑇 (𝑋 𝑉 ¬ 𝑋 𝑆) 𝑌 𝑇) → {𝑌} ⊆ 𝑇)
15 ssequn2 3110 . . . 4 ({𝑌} ⊆ 𝑇 ↔ (𝑇 ∪ {𝑌}) = 𝑇)
1614, 15sylib 127 . . 3 ((𝐹:𝑆𝑇 (𝑋 𝑉 ¬ 𝑋 𝑆) 𝑌 𝑇) → (𝑇 ∪ {𝑌}) = 𝑇)
17 feq3 4975 . . 3 ((𝑇 ∪ {𝑌}) = 𝑇 → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌}) ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶𝑇))
1816, 17syl 14 . 2 ((𝐹:𝑆𝑇 (𝑋 𝑉 ¬ 𝑋 𝑆) 𝑌 𝑇) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌}) ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶𝑇))
1912, 18mpbid 135 1 ((𝐹:𝑆𝑇 (𝑋 𝑉 ¬ 𝑋 𝑆) 𝑌 𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶𝑇)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  cun 2909  cin 2910  wss 2911  c0 3218  {csn 3367  cop 3370  wf 4841  1-1-ontowf1o 4844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852
This theorem is referenced by: (None)
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