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Theorem djurf1o 9953
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 9943 . . 3 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2 1onn 8678 . . . . . 6 1o ∈ ω
3 snidg 4660 . . . . . 6 (1o ∈ ω → 1o ∈ {1o})
42, 3ax-mp 5 . . . . 5 1o ∈ {1o}
5 opelxpi 5722 . . . . 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 6921 . . 3 ((⊤ ∧ 𝑦 ∈ ({1o} × V)) → (2nd𝑦) ∈ V)
9 1st2nd2 8053 . . . . . . . 8 (𝑦 ∈ ({1o} × V) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
10 xp1st 8046 . . . . . . . . . 10 (𝑦 ∈ ({1o} × V) → (1st𝑦) ∈ {1o})
11 elsni 4643 . . . . . . . . . 10 ((1st𝑦) ∈ {1o} → (1st𝑦) = 1o)
1210, 11syl 17 . . . . . . . . 9 (𝑦 ∈ ({1o} × V) → (1st𝑦) = 1o)
1312opeq1d 4879 . . . . . . . 8 (𝑦 ∈ ({1o} × V) → ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨1o, (2nd𝑦)⟩)
149, 13eqtrd 2777 . . . . . . 7 (𝑦 ∈ ({1o} × V) → 𝑦 = ⟨1o, (2nd𝑦)⟩)
1514eqeq2d 2748 . . . . . 6 (𝑦 ∈ ({1o} × V) → (⟨1o, 𝑥⟩ = 𝑦 ↔ ⟨1o, 𝑥⟩ = ⟨1o, (2nd𝑦)⟩))
16 eqcom 2744 . . . . . 6 (⟨1o, 𝑥⟩ = 𝑦𝑦 = ⟨1o, 𝑥⟩)
17 eqid 2737 . . . . . . 7 1o = 1o
18 1oex 8516 . . . . . . . 8 1o ∈ V
19 vex 3484 . . . . . . . 8 𝑥 ∈ V
2018, 19opth 5481 . . . . . . 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 1547 1 inr:V–1-1-onto→({1o} × V)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wtru 1541  wcel 2108  Vcvv 3480  {csn 4626  cop 4632   × cxp 5683  1-1-ontowf1o 6560  cfv 6561  ωcom 7887  1st c1st 8012  2nd c2nd 8013  1oc1o 8499  inrcinr 9940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-inr 9943
This theorem is referenced by:  inrresf  9956  inrresf1  9957  djuin  9958  djuun  9966
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