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Theorem fsn 5453
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 5167 . . . . . . . 8 ((𝐹:{𝐴}⟶{𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}))
2 velsn 3458 . . . . . . . . 9 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3 velsn 3458 . . . . . . . . 9 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
42, 3anbi12i 448 . . . . . . . 8 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) ↔ (𝑥 = 𝐴𝑦 = 𝐵))
51, 4sylib 120 . . . . . . 7 ((𝐹:{𝐴}⟶{𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥 = 𝐴𝑦 = 𝐵))
65ex 113 . . . . . 6 (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 = 𝐴𝑦 = 𝐵)))
7 fsn.1 . . . . . . . . . 10 𝐴 ∈ V
87snid 3470 . . . . . . . . 9 𝐴 ∈ {𝐴}
9 feu 5177 . . . . . . . . 9 ((𝐹:{𝐴}⟶{𝐵} ∧ 𝐴 ∈ {𝐴}) → ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
108, 9mpan2 416 . . . . . . . 8 (𝐹:{𝐴}⟶{𝐵} → ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
113anbi1i 446 . . . . . . . . . . 11 ((𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
12 opeq2 3618 . . . . . . . . . . . . . 14 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
1312eleq1d 2156 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
1413pm5.32i 442 . . . . . . . . . . . 12 ((𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (𝑦 = 𝐵 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
15 ancom 262 . . . . . . . . . . . 12 ((⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵) ↔ (𝑦 = 𝐵 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
1614, 15bitr4i 185 . . . . . . . . . . 11 ((𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵))
1711, 16bitr2i 183 . . . . . . . . . 10 ((⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵) ↔ (𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
1817eubii 1957 . . . . . . . . 9 (∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵) ↔ ∃!𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
19 fsn.2 . . . . . . . . . . . 12 𝐵 ∈ V
2019eueq1 2785 . . . . . . . . . . 11 ∃!𝑦 𝑦 = 𝐵
2120biantru 296 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ ∃!𝑦 𝑦 = 𝐵))
22 euanv 2005 . . . . . . . . . 10 (∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ ∃!𝑦 𝑦 = 𝐵))
2321, 22bitr4i 185 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵))
24 df-reu 2366 . . . . . . . . 9 (∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
2518, 23, 243bitr4i 210 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
2610, 25sylibr 132 . . . . . . 7 (𝐹:{𝐴}⟶{𝐵} → ⟨𝐴, 𝐵⟩ ∈ 𝐹)
27 opeq12 3619 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
2827eleq1d 2156 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
2926, 28syl5ibrcom 155 . . . . . 6 (𝐹:{𝐴}⟶{𝐵} → ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝐹))
306, 29impbid 127 . . . . 5 (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (𝑥 = 𝐴𝑦 = 𝐵)))
31 vex 2622 . . . . . . . 8 𝑥 ∈ V
32 vex 2622 . . . . . . . 8 𝑦 ∈ V
3331, 32opex 4047 . . . . . . 7 𝑥, 𝑦⟩ ∈ V
3433elsn 3457 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
357, 19opth2 4058 . . . . . 6 (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴𝑦 = 𝐵))
3634, 35bitr2i 183 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
3730, 36syl6bb 194 . . . 4 (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩}))
3837alrimivv 1803 . . 3 (𝐹:{𝐴}⟶{𝐵} → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩}))
39 frel 5151 . . . 4 (𝐹:{𝐴}⟶{𝐵} → Rel 𝐹)
407, 19relsnop 4532 . . . 4 Rel {⟨𝐴, 𝐵⟩}
41 eqrel 4515 . . . 4 ((Rel 𝐹 ∧ Rel {⟨𝐴, 𝐵⟩}) → (𝐹 = {⟨𝐴, 𝐵⟩} ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})))
4239, 40, 41sylancl 404 . . 3 (𝐹:{𝐴}⟶{𝐵} → (𝐹 = {⟨𝐴, 𝐵⟩} ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})))
4338, 42mpbird 165 . 2 (𝐹:{𝐴}⟶{𝐵} → 𝐹 = {⟨𝐴, 𝐵⟩})
447, 19f1osn 5277 . . . 4 {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}
45 f1oeq1 5228 . . . 4 (𝐹 = {⟨𝐴, 𝐵⟩} → (𝐹:{𝐴}–1-1-onto→{𝐵} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}))
4644, 45mpbiri 166 . . 3 (𝐹 = {⟨𝐴, 𝐵⟩} → 𝐹:{𝐴}–1-1-onto→{𝐵})
47 f1of 5237 . . 3 (𝐹:{𝐴}–1-1-onto→{𝐵} → 𝐹:{𝐴}⟶{𝐵})
4846, 47syl 14 . 2 (𝐹 = {⟨𝐴, 𝐵⟩} → 𝐹:{𝐴}⟶{𝐵})
4943, 48impbii 124 1 (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩})
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wal 1287   = wceq 1289  wcel 1438  ∃!weu 1948  ∃!wreu 2361  Vcvv 2619  {csn 3441  cop 3444  Rel wrel 4433  wf 4998  1-1-ontowf1o 5001
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009
This theorem is referenced by:  fsng  5454  mapsn  6427
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