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

Theorem fsn 5599
Description: A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 10-Dec-2003.)
Hypotheses
Ref Expression
fsn.1 𝐴 ∈ V
fsn.2 𝐵 ∈ V
Assertion
Ref Expression
fsn (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩})

Proof of Theorem fsn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opelf 5301 . . . . . . . 8 ((𝐹:{𝐴}⟶{𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}))
2 velsn 3548 . . . . . . . . 9 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3 velsn 3548 . . . . . . . . 9 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
42, 3anbi12i 456 . . . . . . . 8 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) ↔ (𝑥 = 𝐴𝑦 = 𝐵))
51, 4sylib 121 . . . . . . 7 ((𝐹:{𝐴}⟶{𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥 = 𝐴𝑦 = 𝐵))
65ex 114 . . . . . 6 (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 = 𝐴𝑦 = 𝐵)))
7 fsn.1 . . . . . . . . . 10 𝐴 ∈ V
87snid 3562 . . . . . . . . 9 𝐴 ∈ {𝐴}
9 feu 5312 . . . . . . . . 9 ((𝐹:{𝐴}⟶{𝐵} ∧ 𝐴 ∈ {𝐴}) → ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
108, 9mpan2 422 . . . . . . . 8 (𝐹:{𝐴}⟶{𝐵} → ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
113anbi1i 454 . . . . . . . . . . 11 ((𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
12 opeq2 3713 . . . . . . . . . . . . . 14 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
1312eleq1d 2209 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
1413pm5.32i 450 . . . . . . . . . . . 12 ((𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (𝑦 = 𝐵 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
15 ancom 264 . . . . . . . . . . . 12 ((⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵) ↔ (𝑦 = 𝐵 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
1614, 15bitr4i 186 . . . . . . . . . . 11 ((𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵))
1711, 16bitr2i 184 . . . . . . . . . 10 ((⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵) ↔ (𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
1817eubii 2009 . . . . . . . . 9 (∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵) ↔ ∃!𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
19 fsn.2 . . . . . . . . . . . 12 𝐵 ∈ V
2019eueq1 2859 . . . . . . . . . . 11 ∃!𝑦 𝑦 = 𝐵
2120biantru 300 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ ∃!𝑦 𝑦 = 𝐵))
22 euanv 2057 . . . . . . . . . 10 (∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ ∃!𝑦 𝑦 = 𝐵))
2321, 22bitr4i 186 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵))
24 df-reu 2424 . . . . . . . . 9 (∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
2518, 23, 243bitr4i 211 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
2610, 25sylibr 133 . . . . . . 7 (𝐹:{𝐴}⟶{𝐵} → ⟨𝐴, 𝐵⟩ ∈ 𝐹)
27 opeq12 3714 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
2827eleq1d 2209 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
2926, 28syl5ibrcom 156 . . . . . 6 (𝐹:{𝐴}⟶{𝐵} → ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝐹))
306, 29impbid 128 . . . . 5 (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (𝑥 = 𝐴𝑦 = 𝐵)))
31 vex 2692 . . . . . . . 8 𝑥 ∈ V
32 vex 2692 . . . . . . . 8 𝑦 ∈ V
3331, 32opex 4158 . . . . . . 7 𝑥, 𝑦⟩ ∈ V
3433elsn 3547 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
357, 19opth2 4169 . . . . . 6 (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴𝑦 = 𝐵))
3634, 35bitr2i 184 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
3730, 36syl6bb 195 . . . 4 (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩}))
3837alrimivv 1848 . . 3 (𝐹:{𝐴}⟶{𝐵} → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩}))
39 frel 5284 . . . 4 (𝐹:{𝐴}⟶{𝐵} → Rel 𝐹)
407, 19relsnop 4652 . . . 4 Rel {⟨𝐴, 𝐵⟩}
41 eqrel 4635 . . . 4 ((Rel 𝐹 ∧ Rel {⟨𝐴, 𝐵⟩}) → (𝐹 = {⟨𝐴, 𝐵⟩} ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})))
4239, 40, 41sylancl 410 . . 3 (𝐹:{𝐴}⟶{𝐵} → (𝐹 = {⟨𝐴, 𝐵⟩} ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})))
4338, 42mpbird 166 . 2 (𝐹:{𝐴}⟶{𝐵} → 𝐹 = {⟨𝐴, 𝐵⟩})
447, 19f1osn 5414 . . . 4 {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}
45 f1oeq1 5363 . . . 4 (𝐹 = {⟨𝐴, 𝐵⟩} → (𝐹:{𝐴}–1-1-onto→{𝐵} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}))
4644, 45mpbiri 167 . . 3 (𝐹 = {⟨𝐴, 𝐵⟩} → 𝐹:{𝐴}–1-1-onto→{𝐵})
47 f1of 5374 . . 3 (𝐹:{𝐴}–1-1-onto→{𝐵} → 𝐹:{𝐴}⟶{𝐵})
4846, 47syl 14 . 2 (𝐹 = {⟨𝐴, 𝐵⟩} → 𝐹:{𝐴}⟶{𝐵})
4943, 48impbii 125 1 (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩})
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wal 1330   = wceq 1332  wcel 1481  ∃!weu 2000  ∃!wreu 2419  Vcvv 2689  {csn 3531  cop 3534  Rel wrel 4551  wf 5126  1-1-ontowf1o 5129
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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137
This theorem is referenced by:  fsng  5600  mapsn  6591
  Copyright terms: Public domain W3C validator