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Theorem f1o2sn 6314
 Description: A singleton with a nested ordered pair is a 1-1 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 4859 . . 3 𝐸, 𝐸⟩ ∈ V
2 simpr 476 . . 3 ((𝐸𝑉𝑋𝑊) → 𝑋𝑊)
3 f1osng 6089 . . 3 ((⟨𝐸, 𝐸⟩ ∈ V ∧ 𝑋𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:{⟨𝐸, 𝐸⟩}–1-1-onto→{𝑋})
41, 2, 3sylancr 694 . 2 ((𝐸𝑉𝑋𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:{⟨𝐸, 𝐸⟩}–1-1-onto→{𝑋})
5 xpsng 6312 . . . . . 6 ((𝐸𝑉𝐸𝑉) → ({𝐸} × {𝐸}) = {⟨𝐸, 𝐸⟩})
65anidms 675 . . . . 5 (𝐸𝑉 → ({𝐸} × {𝐸}) = {⟨𝐸, 𝐸⟩})
76eqcomd 2616 . . . 4 (𝐸𝑉 → {⟨𝐸, 𝐸⟩} = ({𝐸} × {𝐸}))
87adantr 480 . . 3 ((𝐸𝑉𝑋𝑊) → {⟨𝐸, 𝐸⟩} = ({𝐸} × {𝐸}))
9 f1oeq2 6041 . . 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 221 1 ((𝐸𝑉𝑋𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:({𝐸} × {𝐸})–1-1-onto→{𝑋})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ⟨cop 4131   × cxp 5036  –1-1-onto→wf1o 5803 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811 This theorem is referenced by:  mat1dimelbas  20096
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