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Theorem f1osng 6144
Description: A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.)
Assertion
Ref Expression
f1osng ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})

Proof of Theorem f1osng
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sneq 4165 . . . 4 (𝑎 = 𝐴 → {𝑎} = {𝐴})
2 f1oeq2 6095 . . . 4 ({𝑎} = {𝐴} → ({⟨𝑎, 𝑏⟩}:{𝑎}–1-1-onto→{𝑏} ↔ {⟨𝑎, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏}))
31, 2syl 17 . . 3 (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}:{𝑎}–1-1-onto→{𝑏} ↔ {⟨𝑎, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏}))
4 opeq1 4377 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
54sneqd 4167 . . . 4 (𝑎 = 𝐴 → {⟨𝑎, 𝑏⟩} = {⟨𝐴, 𝑏⟩})
6 f1oeq1 6094 . . . 4 ({⟨𝑎, 𝑏⟩} = {⟨𝐴, 𝑏⟩} → ({⟨𝑎, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏}))
75, 6syl 17 . . 3 (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏}))
83, 7bitrd 268 . 2 (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}:{𝑎}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏}))
9 sneq 4165 . . . 4 (𝑏 = 𝐵 → {𝑏} = {𝐵})
10 f1oeq3 6096 . . . 4 ({𝑏} = {𝐵} → ({⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝐵}))
119, 10syl 17 . . 3 (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝐵}))
12 opeq2 4378 . . . . 5 (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
1312sneqd 4167 . . . 4 (𝑏 = 𝐵 → {⟨𝐴, 𝑏⟩} = {⟨𝐴, 𝐵⟩})
14 f1oeq1 6094 . . . 4 ({⟨𝐴, 𝑏⟩} = {⟨𝐴, 𝐵⟩} → ({⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝐵} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}))
1513, 14syl 17 . . 3 (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝐵} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}))
1611, 15bitrd 268 . 2 (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}))
17 vex 3193 . . 3 𝑎 ∈ V
18 vex 3193 . . 3 𝑏 ∈ V
1917, 18f1osn 6143 . 2 {⟨𝑎, 𝑏⟩}:{𝑎}–1-1-onto→{𝑏}
208, 16, 19vtocl2g 3260 1 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {csn 4155  cop 4161  1-1-ontowf1o 5856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864
This theorem is referenced by:  f1sng  6145  f1oprswap  6147  f1oprg  6148  f1o2sn  6373  fsnunf  6416  fsnex  6503  suppsnop  7269  ralxpmap  7867  enfixsn  8029  fseqenlem1  8807  canthp1lem2  9435  sumsnf  14422  sumsn  14424  prodsn  14636  prodsnf  14638  vdwlem8  15635  gsumws1  17316  symg1bas  17756  dprdsn  18375  eupthp1  26976  poimirlem16  33096  poimirlem17  33097  poimirlem19  33099  poimirlem20  33100  mapfzcons  36798  sumsnd  38707  mapsnd  38897  1hegrlfgr  41031
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