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Theorem f1o2sn 6896
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 5347 . . 3 𝐸, 𝐸⟩ ∈ V
2 simpr 485 . . 3 ((𝐸𝑉𝑋𝑊) → 𝑋𝑊)
3 f1osng 6648 . . 3 ((⟨𝐸, 𝐸⟩ ∈ V ∧ 𝑋𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:{⟨𝐸, 𝐸⟩}–1-1-onto→{𝑋})
41, 2, 3sylancr 587 . 2 ((𝐸𝑉𝑋𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:{⟨𝐸, 𝐸⟩}–1-1-onto→{𝑋})
5 xpsng 6893 . . . . . 6 ((𝐸𝑉𝐸𝑉) → ({𝐸} × {𝐸}) = {⟨𝐸, 𝐸⟩})
65anidms 567 . . . . 5 (𝐸𝑉 → ({𝐸} × {𝐸}) = {⟨𝐸, 𝐸⟩})
76eqcomd 2824 . . . 4 (𝐸𝑉 → {⟨𝐸, 𝐸⟩} = ({𝐸} × {𝐸}))
87adantr 481 . . 3 ((𝐸𝑉𝑋𝑊) → {⟨𝐸, 𝐸⟩} = ({𝐸} × {𝐸}))
98f1oeq2d 6604 . 2 ((𝐸𝑉𝑋𝑊) → ({⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:{⟨𝐸, 𝐸⟩}–1-1-onto→{𝑋} ↔ {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:({𝐸} × {𝐸})–1-1-onto→{𝑋}))
104, 9mpbid 233 1 ((𝐸𝑉𝑋𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:({𝐸} × {𝐸})–1-1-onto→{𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  {csn 4557  cop 4563   × cxp 5546  1-1-ontowf1o 6347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355
This theorem is referenced by:  mat1dimelbas  21008
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