MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fsnunf Structured version   Visualization version   GIF version

Theorem fsnunf 6334
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 1053 . . 3 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → 𝐹:𝑆𝑇)
2 simp2l 1079 . . . . 5 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → 𝑋𝑉)
3 simp3 1055 . . . . 5 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → 𝑌𝑇)
4 f1osng 6074 . . . . 5 ((𝑋𝑉𝑌𝑇) → {⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌})
52, 3, 4syl2anc 690 . . . 4 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → {⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌})
6 f1of 6035 . . . 4 ({⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌} → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
75, 6syl 17 . . 3 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
8 simp2r 1080 . . . 4 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → ¬ 𝑋𝑆)
9 disjsn 4191 . . . 4 ((𝑆 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋𝑆)
108, 9sylibr 222 . . 3 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → (𝑆 ∩ {𝑋}) = ∅)
11 fun 5965 . . 3 (((𝐹:𝑆𝑇 ∧ {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌}) ∧ (𝑆 ∩ {𝑋}) = ∅) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌}))
121, 7, 10, 11syl21anc 1316 . 2 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌}))
13 snssi 4279 . . . . 5 (𝑌𝑇 → {𝑌} ⊆ 𝑇)
14133ad2ant3 1076 . . . 4 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → {𝑌} ⊆ 𝑇)
15 ssequn2 3747 . . . 4 ({𝑌} ⊆ 𝑇 ↔ (𝑇 ∪ {𝑌}) = 𝑇)
1614, 15sylib 206 . . 3 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → (𝑇 ∪ {𝑌}) = 𝑇)
1716feq3d 5931 . 2 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌}) ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶𝑇))
1812, 17mpbid 220 1 ((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  cun 3537  cin 3538  wss 3539  c0 3873  {csn 4124  cop 4130  wf 5786  1-1-ontowf1o 5789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797
This theorem is referenced by:  fsnunf2  6335  fnchoice  38014  nnsum4primeseven  40021  nnsum4primesevenALTV  40022
  Copyright terms: Public domain W3C validator