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Theorem fsn 7082
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 6704 . . . . . . . 8 ((𝐹:{𝐴}⟶{𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}))
2 velsn 4603 . . . . . . . . 9 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3 velsn 4603 . . . . . . . . 9 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
42, 3anbi12i 628 . . . . . . . 8 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) ↔ (𝑥 = 𝐴𝑦 = 𝐵))
51, 4sylib 217 . . . . . . 7 ((𝐹:{𝐴}⟶{𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥 = 𝐴𝑦 = 𝐵))
65ex 414 . . . . . 6 (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 = 𝐴𝑦 = 𝐵)))
7 fsn.1 . . . . . . . . . 10 𝐴 ∈ V
87snid 4623 . . . . . . . . 9 𝐴 ∈ {𝐴}
9 feu 6719 . . . . . . . . 9 ((𝐹:{𝐴}⟶{𝐵} ∧ 𝐴 ∈ {𝐴}) → ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
108, 9mpan2 690 . . . . . . . 8 (𝐹:{𝐴}⟶{𝐵} → ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
113anbi1i 625 . . . . . . . . . . 11 ((𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
12 opeq2 4832 . . . . . . . . . . . . . 14 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
1312eleq1d 2819 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
1413pm5.32i 576 . . . . . . . . . . . 12 ((𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (𝑦 = 𝐵 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
1514biancomi 464 . . . . . . . . . . 11 ((𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵))
1611, 15bitr2i 276 . . . . . . . . . 10 ((⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵) ↔ (𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
1716eubii 2580 . . . . . . . . 9 (∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵) ↔ ∃!𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
18 fsn.2 . . . . . . . . . . . 12 𝐵 ∈ V
1918eueqi 3668 . . . . . . . . . . 11 ∃!𝑦 𝑦 = 𝐵
2019biantru 531 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ ∃!𝑦 𝑦 = 𝐵))
21 euanv 2621 . . . . . . . . . 10 (∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ ∃!𝑦 𝑦 = 𝐵))
2220, 21bitr4i 278 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹𝑦 = 𝐵))
23 df-reu 3353 . . . . . . . . 9 (∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
2417, 22, 233bitr4i 303 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
2510, 24sylibr 233 . . . . . . 7 (𝐹:{𝐴}⟶{𝐵} → ⟨𝐴, 𝐵⟩ ∈ 𝐹)
26 opeq12 4833 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
2726eleq1d 2819 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
2825, 27syl5ibrcom 247 . . . . . 6 (𝐹:{𝐴}⟶{𝐵} → ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝐹))
296, 28impbid 211 . . . . 5 (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (𝑥 = 𝐴𝑦 = 𝐵)))
30 opex 5422 . . . . . . 7 𝑥, 𝑦⟩ ∈ V
3130elsn 4602 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
327, 18opth2 5438 . . . . . 6 (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴𝑦 = 𝐵))
3331, 32bitr2i 276 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
3429, 33bitrdi 287 . . . 4 (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩}))
3534alrimivv 1932 . . 3 (𝐹:{𝐴}⟶{𝐵} → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩}))
36 frel 6674 . . . 4 (𝐹:{𝐴}⟶{𝐵} → Rel 𝐹)
377, 18relsnop 5762 . . . 4 Rel {⟨𝐴, 𝐵⟩}
38 eqrel 5741 . . . 4 ((Rel 𝐹 ∧ Rel {⟨𝐴, 𝐵⟩}) → (𝐹 = {⟨𝐴, 𝐵⟩} ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})))
3936, 37, 38sylancl 587 . . 3 (𝐹:{𝐴}⟶{𝐵} → (𝐹 = {⟨𝐴, 𝐵⟩} ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})))
4035, 39mpbird 257 . 2 (𝐹:{𝐴}⟶{𝐵} → 𝐹 = {⟨𝐴, 𝐵⟩})
417, 18f1osn 6825 . . . 4 {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}
42 f1oeq1 6773 . . . 4 (𝐹 = {⟨𝐴, 𝐵⟩} → (𝐹:{𝐴}–1-1-onto→{𝐵} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}))
4341, 42mpbiri 258 . . 3 (𝐹 = {⟨𝐴, 𝐵⟩} → 𝐹:{𝐴}–1-1-onto→{𝐵})
44 f1of 6785 . . 3 (𝐹:{𝐴}–1-1-onto→{𝐵} → 𝐹:{𝐴}⟶{𝐵})
4543, 44syl 17 . 2 (𝐹 = {⟨𝐴, 𝐵⟩} → 𝐹:{𝐴}⟶{𝐵})
4640, 45impbii 208 1 (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩})
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wal 1540   = wceq 1542  wcel 2107  ∃!weu 2563  ∃!wreu 3350  Vcvv 3444  {csn 4587  cop 4593  Rel wrel 5639  wf 6493  1-1-ontowf1o 6496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504
This theorem is referenced by:  fsn2  7083  fsng  7084  axlowdimlem7  27939  poimirlem3  36127  poimirlem9  36133  fdc  36250
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