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Theorem f1o2sn 6448
Description: A singleton with a nested ordered pair is a one-to-one function of the cartesian product of two singleton onto a singleton. (Contributed by AV, 15-Aug-2019.)
Assertion
Ref Expression
f1o2sn ((𝐸𝑉𝑋𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:({𝐸} × {𝐸})–1-1-onto→{𝑋})

Proof of Theorem f1o2sn
StepHypRef Expression
1 opex 4962 . . 3 𝐸, 𝐸⟩ ∈ V
2 simpr 476 . . 3 ((𝐸𝑉𝑋𝑊) → 𝑋𝑊)
3 f1osng 6215 . . 3 ((⟨𝐸, 𝐸⟩ ∈ V ∧ 𝑋𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:{⟨𝐸, 𝐸⟩}–1-1-onto→{𝑋})
41, 2, 3sylancr 696 . 2 ((𝐸𝑉𝑋𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:{⟨𝐸, 𝐸⟩}–1-1-onto→{𝑋})
5 xpsng 6446 . . . . . 6 ((𝐸𝑉𝐸𝑉) → ({𝐸} × {𝐸}) = {⟨𝐸, 𝐸⟩})
65anidms 678 . . . . 5 (𝐸𝑉 → ({𝐸} × {𝐸}) = {⟨𝐸, 𝐸⟩})
76eqcomd 2657 . . . 4 (𝐸𝑉 → {⟨𝐸, 𝐸⟩} = ({𝐸} × {𝐸}))
87adantr 480 . . 3 ((𝐸𝑉𝑋𝑊) → {⟨𝐸, 𝐸⟩} = ({𝐸} × {𝐸}))
9 f1oeq2 6166 . . 3 ({⟨𝐸, 𝐸⟩} = ({𝐸} × {𝐸}) → ({⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:{⟨𝐸, 𝐸⟩}–1-1-onto→{𝑋} ↔ {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:({𝐸} × {𝐸})–1-1-onto→{𝑋}))
108, 9syl 17 . 2 ((𝐸𝑉𝑋𝑊) → ({⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:{⟨𝐸, 𝐸⟩}–1-1-onto→{𝑋} ↔ {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:({𝐸} × {𝐸})–1-1-onto→{𝑋}))
114, 10mpbid 222 1 ((𝐸𝑉𝑋𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:({𝐸} × {𝐸})–1-1-onto→{𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  {csn 4210  cop 4216   × cxp 5141  1-1-ontowf1o 5925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933
This theorem is referenced by:  mat1dimelbas  20325
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