Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  fsnunf GIF version

Theorem fsnunf 5632
 Description: Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Assertion
Ref Expression
fsnunf ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶𝑇)

Proof of Theorem fsnunf
StepHypRef Expression
1 simp1 982 . . 3 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → 𝐹:𝑆𝑇)
2 simp2l 1008 . . . . 5 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → 𝑋𝑉)
3 simp3 984 . . . . 5 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → 𝑌𝑇)
4 f1osng 5420 . . . . 5 ((𝑋𝑉𝑌𝑇) → {⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌})
52, 3, 4syl2anc 409 . . . 4 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → {⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌})
6 f1of 5379 . . . 4 ({⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌} → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
75, 6syl 14 . . 3 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
8 simp2r 1009 . . . 4 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → ¬ 𝑋𝑆)
9 disjsn 3595 . . . 4 ((𝑆 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋𝑆)
108, 9sylibr 133 . . 3 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → (𝑆 ∩ {𝑋}) = ∅)
11 fun 5307 . . 3 (((𝐹:𝑆𝑇 ∧ {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌}) ∧ (𝑆 ∩ {𝑋}) = ∅) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌}))
121, 7, 10, 11syl21anc 1216 . 2 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌}))
13 snssi 3674 . . . . 5 (𝑌𝑇 → {𝑌} ⊆ 𝑇)
14133ad2ant3 1005 . . . 4 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → {𝑌} ⊆ 𝑇)
15 ssequn2 3256 . . . 4 ({𝑌} ⊆ 𝑇 ↔ (𝑇 ∪ {𝑌}) = 𝑇)
1614, 15sylib 121 . . 3 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → (𝑇 ∪ {𝑌}) = 𝑇)
17 feq3 5269 . . 3 ((𝑇 ∪ {𝑌}) = 𝑇 → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌}) ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶𝑇))
1816, 17syl 14 . 2 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌}) ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶𝑇))
1912, 18mpbid 146 1 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶𝑇)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 963   = wceq 1332   ∈ wcel 2112   ∪ cun 3076   ∩ cin 3077   ⊆ wss 3078  ∅c0 3370  {csn 3534  ⟨cop 3537  ⟶wf 5131  –1-1-onto→wf1o 5134 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2115  ax-ext 2123  ax-sep 4056  ax-pow 4108  ax-pr 4142 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1732  df-eu 1993  df-mo 1994  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-rex 2424  df-v 2693  df-dif 3080  df-un 3082  df-in 3084  df-ss 3091  df-nul 3371  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-br 3940  df-opab 4000  df-id 4226  df-xp 4557  df-rel 4558  df-cnv 4559  df-co 4560  df-dm 4561  df-rn 4562  df-fun 5137  df-fn 5138  df-f 5139  df-f1 5140  df-fo 5141  df-f1o 5142 This theorem is referenced by:  tfrcllemsucfn  6262  ennnfonelemg  11988
 Copyright terms: Public domain W3C validator