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Theorem djurf1o 9951
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 9941 . . 3 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2 1onn 8677 . . . . . 6 1o ∈ ω
3 snidg 4665 . . . . . 6 (1o ∈ ω → 1o ∈ {1o})
42, 3ax-mp 5 . . . . 5 1o ∈ {1o}
5 opelxpi 5726 . . . . 5 ((1o ∈ {1o} ∧ 𝑥 ∈ V) → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
64, 5mpan 690 . . . 4 (𝑥 ∈ V → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
76adantl 481 . . 3 ((⊤ ∧ 𝑥 ∈ V) → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
8 fvexd 6922 . . 3 ((⊤ ∧ 𝑦 ∈ ({1o} × V)) → (2nd𝑦) ∈ V)
9 1st2nd2 8052 . . . . . . . 8 (𝑦 ∈ ({1o} × V) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
10 xp1st 8045 . . . . . . . . . 10 (𝑦 ∈ ({1o} × V) → (1st𝑦) ∈ {1o})
11 elsni 4648 . . . . . . . . . 10 ((1st𝑦) ∈ {1o} → (1st𝑦) = 1o)
1210, 11syl 17 . . . . . . . . 9 (𝑦 ∈ ({1o} × V) → (1st𝑦) = 1o)
1312opeq1d 4884 . . . . . . . 8 (𝑦 ∈ ({1o} × V) → ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨1o, (2nd𝑦)⟩)
149, 13eqtrd 2775 . . . . . . 7 (𝑦 ∈ ({1o} × V) → 𝑦 = ⟨1o, (2nd𝑦)⟩)
1514eqeq2d 2746 . . . . . 6 (𝑦 ∈ ({1o} × V) → (⟨1o, 𝑥⟩ = 𝑦 ↔ ⟨1o, 𝑥⟩ = ⟨1o, (2nd𝑦)⟩))
16 eqcom 2742 . . . . . 6 (⟨1o, 𝑥⟩ = 𝑦𝑦 = ⟨1o, 𝑥⟩)
17 eqid 2735 . . . . . . 7 1o = 1o
18 1oex 8515 . . . . . . . 8 1o ∈ V
19 vex 3482 . . . . . . . 8 𝑥 ∈ V
2018, 19opth 5487 . . . . . . 7 (⟨1o, 𝑥⟩ = ⟨1o, (2nd𝑦)⟩ ↔ (1o = 1o𝑥 = (2nd𝑦)))
2117, 20mpbiran 709 . . . . . 6 (⟨1o, 𝑥⟩ = ⟨1o, (2nd𝑦)⟩ ↔ 𝑥 = (2nd𝑦))
2215, 16, 213bitr3g 313 . . . . 5 (𝑦 ∈ ({1o} × V) → (𝑦 = ⟨1o, 𝑥⟩ ↔ 𝑥 = (2nd𝑦)))
2322bicomd 223 . . . 4 (𝑦 ∈ ({1o} × V) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨1o, 𝑥⟩))
2423ad2antll 729 . . 3 ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({1o} × V))) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨1o, 𝑥⟩))
251, 7, 8, 24f1o2d 7687 . 2 (⊤ → inr:V–1-1-onto→({1o} × V))
2625mptru 1544 1 inr:V–1-1-onto→({1o} × V)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wtru 1538  wcel 2106  Vcvv 3478  {csn 4631  cop 4637   × cxp 5687  1-1-ontowf1o 6562  cfv 6563  ωcom 7887  1st c1st 8011  2nd c2nd 8012  1oc1o 8498  inrcinr 9938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-inr 9941
This theorem is referenced by:  inrresf  9954  inrresf1  9955  djuin  9956  djuun  9964
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