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Theorem f1o2sn 6551
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 5060 . . 3 𝐸, 𝐸⟩ ∈ V
2 simpr 471 . . 3 ((𝐸𝑉𝑋𝑊) → 𝑋𝑊)
3 f1osng 6318 . . 3 ((⟨𝐸, 𝐸⟩ ∈ V ∧ 𝑋𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:{⟨𝐸, 𝐸⟩}–1-1-onto→{𝑋})
41, 2, 3sylancr 575 . 2 ((𝐸𝑉𝑋𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:{⟨𝐸, 𝐸⟩}–1-1-onto→{𝑋})
5 xpsng 6549 . . . . . 6 ((𝐸𝑉𝐸𝑉) → ({𝐸} × {𝐸}) = {⟨𝐸, 𝐸⟩})
65anidms 556 . . . . 5 (𝐸𝑉 → ({𝐸} × {𝐸}) = {⟨𝐸, 𝐸⟩})
76eqcomd 2777 . . . 4 (𝐸𝑉 → {⟨𝐸, 𝐸⟩} = ({𝐸} × {𝐸}))
87adantr 466 . . 3 ((𝐸𝑉𝑋𝑊) → {⟨𝐸, 𝐸⟩} = ({𝐸} × {𝐸}))
9 f1oeq2 6269 . . 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 382   = wceq 1631  wcel 2145  Vcvv 3351  {csn 4316  cop 4322   × cxp 5247  1-1-ontowf1o 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038
This theorem is referenced by:  mat1dimelbas  20495
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