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Theorem djurf1o 8938
Description: The right injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)
Assertion
Ref Expression
djurf1o inr:V–1-1-onto→({1𝑜} × V)

Proof of Theorem djurf1o
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-inr 8929 . . 3 inr = (𝑥 ∈ V ↦ ⟨1𝑜, 𝑥⟩)
2 1onn 7872 . . . . . 6 1𝑜 ∈ ω
3 snidg 4343 . . . . . 6 (1𝑜 ∈ ω → 1𝑜 ∈ {1𝑜})
42, 3ax-mp 5 . . . . 5 1𝑜 ∈ {1𝑜}
5 opelxpi 5288 . . . . 5 ((1𝑜 ∈ {1𝑜} ∧ 𝑥 ∈ V) → ⟨1𝑜, 𝑥⟩ ∈ ({1𝑜} × V))
64, 5mpan 662 . . . 4 (𝑥 ∈ V → ⟨1𝑜, 𝑥⟩ ∈ ({1𝑜} × V))
76adantl 467 . . 3 ((⊤ ∧ 𝑥 ∈ V) → ⟨1𝑜, 𝑥⟩ ∈ ({1𝑜} × V))
8 xp2nd 7347 . . . 4 (𝑦 ∈ ({1𝑜} × V) → (2nd𝑦) ∈ V)
98adantl 467 . . 3 ((⊤ ∧ 𝑦 ∈ ({1𝑜} × V)) → (2nd𝑦) ∈ V)
10 1st2nd2 7353 . . . . . . . 8 (𝑦 ∈ ({1𝑜} × V) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
11 xp1st 7346 . . . . . . . . . 10 (𝑦 ∈ ({1𝑜} × V) → (1st𝑦) ∈ {1𝑜})
12 elsni 4331 . . . . . . . . . 10 ((1st𝑦) ∈ {1𝑜} → (1st𝑦) = 1𝑜)
1311, 12syl 17 . . . . . . . . 9 (𝑦 ∈ ({1𝑜} × V) → (1st𝑦) = 1𝑜)
1413opeq1d 4543 . . . . . . . 8 (𝑦 ∈ ({1𝑜} × V) → ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨1𝑜, (2nd𝑦)⟩)
1510, 14eqtrd 2804 . . . . . . 7 (𝑦 ∈ ({1𝑜} × V) → 𝑦 = ⟨1𝑜, (2nd𝑦)⟩)
1615eqeq2d 2780 . . . . . 6 (𝑦 ∈ ({1𝑜} × V) → (⟨1𝑜, 𝑥⟩ = 𝑦 ↔ ⟨1𝑜, 𝑥⟩ = ⟨1𝑜, (2nd𝑦)⟩))
17 eqcom 2777 . . . . . 6 (⟨1𝑜, 𝑥⟩ = 𝑦𝑦 = ⟨1𝑜, 𝑥⟩)
18 eqid 2770 . . . . . . 7 1𝑜 = 1𝑜
192elexi 3362 . . . . . . . 8 1𝑜 ∈ V
20 vex 3352 . . . . . . . 8 𝑥 ∈ V
2119, 20opth 5072 . . . . . . 7 (⟨1𝑜, 𝑥⟩ = ⟨1𝑜, (2nd𝑦)⟩ ↔ (1𝑜 = 1𝑜𝑥 = (2nd𝑦)))
2218, 21mpbiran 680 . . . . . 6 (⟨1𝑜, 𝑥⟩ = ⟨1𝑜, (2nd𝑦)⟩ ↔ 𝑥 = (2nd𝑦))
2316, 17, 223bitr3g 302 . . . . 5 (𝑦 ∈ ({1𝑜} × V) → (𝑦 = ⟨1𝑜, 𝑥⟩ ↔ 𝑥 = (2nd𝑦)))
2423bicomd 213 . . . 4 (𝑦 ∈ ({1𝑜} × V) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨1𝑜, 𝑥⟩))
2524ad2antll 700 . . 3 ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({1𝑜} × V))) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨1𝑜, 𝑥⟩))
261, 7, 9, 25f1o2d 7033 . 2 (⊤ → inr:V–1-1-onto→({1𝑜} × V))
2726trud 1640 1 inr:V–1-1-onto→({1𝑜} × V)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1630  wtru 1631  wcel 2144  Vcvv 3349  {csn 4314  cop 4320   × cxp 5247  1-1-ontowf1o 6030  cfv 6031  ωcom 7211  1st c1st 7312  2nd c2nd 7313  1𝑜c1o 7705  inrcinr 8926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7212  df-1st 7314  df-2nd 7315  df-1o 7712  df-inr 8929
This theorem is referenced by:  inrresf  8941  inrresf1  8942  djuin  8943  djuun  8951
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