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

Theorem fsn 5680
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 5379 . . . . . . . 8 ((𝐹:{𝐴}⟶{𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}))
2 velsn 3606 . . . . . . . . 9 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3 velsn 3606 . . . . . . . . 9 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
42, 3anbi12i 460 . . . . . . . 8 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) ↔ (𝑥 = 𝐴𝑦 = 𝐵))
51, 4sylib 122 . . . . . . 7 ((𝐹:{𝐴}⟶{𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥 = 𝐴𝑦 = 𝐵))
65ex 115 . . . . . 6 (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 = 𝐴𝑦 = 𝐵)))
7 fsn.1 . . . . . . . . . 10 𝐴 ∈ V
87snid 3620 . . . . . . . . 9 𝐴 ∈ {𝐴}
9 feu 5390 . . . . . . . . 9 ((𝐹:{𝐴}⟶{𝐵} ∧ 𝐴 ∈ {𝐴}) → ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
108, 9mpan2 425 . . . . . . . 8 (𝐹:{𝐴}⟶{𝐵} → ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
113anbi1i 458 . . . . . . . . . . 11 ((𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
12 opeq2 3775 . . . . . . . . . . . . . 14 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
1312eleq1d 2244 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
1413pm5.32i 454 . . . . . . . . . . . 12 ((𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (𝑦 = 𝐵 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
15 ancom 266 . . . . . . . . . . . 12 ((⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵) ↔ (𝑦 = 𝐵 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
1614, 15bitr4i 187 . . . . . . . . . . 11 ((𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵))
1711, 16bitr2i 185 . . . . . . . . . 10 ((⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵) ↔ (𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
1817eubii 2033 . . . . . . . . 9 (∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵) ↔ ∃!𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
19 fsn.2 . . . . . . . . . . . 12 𝐵 ∈ V
2019eueq1 2907 . . . . . . . . . . 11 ∃!𝑦 𝑦 = 𝐵
2120biantru 302 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ ∃!𝑦 𝑦 = 𝐵))
22 euanv 2081 . . . . . . . . . 10 (∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ ∃!𝑦 𝑦 = 𝐵))
2321, 22bitr4i 187 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵))
24 df-reu 2460 . . . . . . . . 9 (∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
2518, 23, 243bitr4i 212 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
2610, 25sylibr 134 . . . . . . 7 (𝐹:{𝐴}⟶{𝐵} → ⟨𝐴, 𝐵⟩ ∈ 𝐹)
27 opeq12 3776 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
2827eleq1d 2244 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
2926, 28syl5ibrcom 157 . . . . . 6 (𝐹:{𝐴}⟶{𝐵} → ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝐹))
306, 29impbid 129 . . . . 5 (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (𝑥 = 𝐴𝑦 = 𝐵)))
31 vex 2738 . . . . . . . 8 𝑥 ∈ V
32 vex 2738 . . . . . . . 8 𝑦 ∈ V
3331, 32opex 4223 . . . . . . 7 𝑥, 𝑦⟩ ∈ V
3433elsn 3605 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
357, 19opth2 4234 . . . . . 6 (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴𝑦 = 𝐵))
3634, 35bitr2i 185 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
3730, 36bitrdi 196 . . . 4 (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩}))
3837alrimivv 1873 . . 3 (𝐹:{𝐴}⟶{𝐵} → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩}))
39 frel 5362 . . . 4 (𝐹:{𝐴}⟶{𝐵} → Rel 𝐹)
407, 19relsnop 4726 . . . 4 Rel {⟨𝐴, 𝐵⟩}
41 eqrel 4709 . . . 4 ((Rel 𝐹 ∧ Rel {⟨𝐴, 𝐵⟩}) → (𝐹 = {⟨𝐴, 𝐵⟩} ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})))
4239, 40, 41sylancl 413 . . 3 (𝐹:{𝐴}⟶{𝐵} → (𝐹 = {⟨𝐴, 𝐵⟩} ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})))
4338, 42mpbird 167 . 2 (𝐹:{𝐴}⟶{𝐵} → 𝐹 = {⟨𝐴, 𝐵⟩})
447, 19f1osn 5493 . . . 4 {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}
45 f1oeq1 5441 . . . 4 (𝐹 = {⟨𝐴, 𝐵⟩} → (𝐹:{𝐴}–1-1-onto→{𝐵} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}))
4644, 45mpbiri 168 . . 3 (𝐹 = {⟨𝐴, 𝐵⟩} → 𝐹:{𝐴}–1-1-onto→{𝐵})
47 f1of 5453 . . 3 (𝐹:{𝐴}–1-1-onto→{𝐵} → 𝐹:{𝐴}⟶{𝐵})
4846, 47syl 14 . 2 (𝐹 = {⟨𝐴, 𝐵⟩} → 𝐹:{𝐴}⟶{𝐵})
4943, 48impbii 126 1 (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩})
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wal 1351   = wceq 1353  ∃!weu 2024  wcel 2146  ∃!wreu 2455  Vcvv 2735  {csn 3589  cop 3592  Rel wrel 4625  wf 5204  1-1-ontowf1o 5207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-reu 2460  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215
This theorem is referenced by:  fsng  5681  mapsn  6680
  Copyright terms: Public domain W3C validator