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Theorem f1osng 6890
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 4641 . . . 4 (𝑎 = 𝐴 → {𝑎} = {𝐴})
21f1oeq2d 6845 . . 3 (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}:{𝑎}–1-1-onto→{𝑏} ↔ {⟨𝑎, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏}))
3 opeq1 4878 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
43sneqd 4643 . . . 4 (𝑎 = 𝐴 → {⟨𝑎, 𝑏⟩} = {⟨𝐴, 𝑏⟩})
54f1oeq1d 6844 . . 3 (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏}))
62, 5bitrd 279 . 2 (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}:{𝑎}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏}))
7 sneq 4641 . . . 4 (𝑏 = 𝐵 → {𝑏} = {𝐵})
87f1oeq3d 6846 . . 3 (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝐵}))
9 opeq2 4879 . . . . 5 (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
109sneqd 4643 . . . 4 (𝑏 = 𝐵 → {⟨𝐴, 𝑏⟩} = {⟨𝐴, 𝐵⟩})
1110f1oeq1d 6844 . . 3 (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝐵} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}))
128, 11bitrd 279 . 2 (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}))
13 vex 3482 . . 3 𝑎 ∈ V
14 vex 3482 . . 3 𝑏 ∈ V
1513, 14f1osn 6889 . 2 {⟨𝑎, 𝑏⟩}:{𝑎}–1-1-onto→{𝑏}
166, 12, 15vtocl2g 3574 1 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {csn 4631  cop 4637  1-1-ontowf1o 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570
This theorem is referenced by:  f1sng  6891  f1oprswap  6893  f1oprg  6894  f1o2sn  7162  fsnunf  7205  fsnex  7303  suppsnop  8202  mapsnd  8925  ralxpmap  8935  en2sn  9080  enfixsn  9120  fseqenlem1  10062  canthp1lem2  10691  sumsnf  15776  prodsn  15995  prodsnf  15997  vdwlem8  17022  gsumws1  18864  symg1bas  19423  dprdsn  20071  eupthp1  30245  s1f1  32912  poimirlem16  37623  poimirlem17  37624  poimirlem19  37626  poimirlem20  37627  metakunt25  42211  mapfzcons  42704  sumsnd  44964  1hegrlfgr  47976
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