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Theorem f1o2sn 6895
 Description: A singleton consisting in a nested ordered pair is a one-to-one function from the cartesian product of two singletons onto a singleton (case where the two singletons are equal). (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 5324 . . 3 𝐸, 𝐸⟩ ∈ V
2 simpr 488 . . 3 ((𝐸𝑉𝑋𝑊) → 𝑋𝑊)
3 f1osng 6642 . . 3 ((⟨𝐸, 𝐸⟩ ∈ V ∧ 𝑋𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:{⟨𝐸, 𝐸⟩}–1-1-onto→{𝑋})
41, 2, 3sylancr 590 . 2 ((𝐸𝑉𝑋𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:{⟨𝐸, 𝐸⟩}–1-1-onto→{𝑋})
5 xpsng 6892 . . . . . 6 ((𝐸𝑉𝐸𝑉) → ({𝐸} × {𝐸}) = {⟨𝐸, 𝐸⟩})
65anidms 570 . . . . 5 (𝐸𝑉 → ({𝐸} × {𝐸}) = {⟨𝐸, 𝐸⟩})
76eqcomd 2764 . . . 4 (𝐸𝑉 → {⟨𝐸, 𝐸⟩} = ({𝐸} × {𝐸}))
87adantr 484 . . 3 ((𝐸𝑉𝑋𝑊) → {⟨𝐸, 𝐸⟩} = ({𝐸} × {𝐸}))
98f1oeq2d 6598 . 2 ((𝐸𝑉𝑋𝑊) → ({⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:{⟨𝐸, 𝐸⟩}–1-1-onto→{𝑋} ↔ {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:({𝐸} × {𝐸})–1-1-onto→{𝑋}))
104, 9mpbid 235 1 ((𝐸𝑉𝑋𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:({𝐸} × {𝐸})–1-1-onto→{𝑋})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409  {csn 4522  ⟨cop 4528   × cxp 5522  –1-1-onto→wf1o 6334 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342 This theorem is referenced by:  mat1dimelbas  21171
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