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Theorem djurf1o 9193
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 9183 . . 3 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2 1onn 8120 . . . . . 6 1o ∈ ω
3 snidg 4508 . . . . . 6 (1o ∈ ω → 1o ∈ {1o})
42, 3ax-mp 5 . . . . 5 1o ∈ {1o}
5 opelxpi 5485 . . . . 5 ((1o ∈ {1o} ∧ 𝑥 ∈ V) → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
64, 5mpan 686 . . . 4 (𝑥 ∈ V → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
76adantl 482 . . 3 ((⊤ ∧ 𝑥 ∈ V) → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
8 fvexd 6558 . . 3 ((⊤ ∧ 𝑦 ∈ ({1o} × V)) → (2nd𝑦) ∈ V)
9 1st2nd2 7589 . . . . . . . 8 (𝑦 ∈ ({1o} × V) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
10 xp1st 7582 . . . . . . . . . 10 (𝑦 ∈ ({1o} × V) → (1st𝑦) ∈ {1o})
11 elsni 4493 . . . . . . . . . 10 ((1st𝑦) ∈ {1o} → (1st𝑦) = 1o)
1210, 11syl 17 . . . . . . . . 9 (𝑦 ∈ ({1o} × V) → (1st𝑦) = 1o)
1312opeq1d 4720 . . . . . . . 8 (𝑦 ∈ ({1o} × V) → ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨1o, (2nd𝑦)⟩)
149, 13eqtrd 2831 . . . . . . 7 (𝑦 ∈ ({1o} × V) → 𝑦 = ⟨1o, (2nd𝑦)⟩)
1514eqeq2d 2805 . . . . . 6 (𝑦 ∈ ({1o} × V) → (⟨1o, 𝑥⟩ = 𝑦 ↔ ⟨1o, 𝑥⟩ = ⟨1o, (2nd𝑦)⟩))
16 eqcom 2802 . . . . . 6 (⟨1o, 𝑥⟩ = 𝑦𝑦 = ⟨1o, 𝑥⟩)
17 eqid 2795 . . . . . . 7 1o = 1o
18 1oex 7966 . . . . . . . 8 1o ∈ V
19 vex 3440 . . . . . . . 8 𝑥 ∈ V
2018, 19opth 5265 . . . . . . 7 (⟨1o, 𝑥⟩ = ⟨1o, (2nd𝑦)⟩ ↔ (1o = 1o𝑥 = (2nd𝑦)))
2117, 20mpbiran 705 . . . . . 6 (⟨1o, 𝑥⟩ = ⟨1o, (2nd𝑦)⟩ ↔ 𝑥 = (2nd𝑦))
2215, 16, 213bitr3g 314 . . . . 5 (𝑦 ∈ ({1o} × V) → (𝑦 = ⟨1o, 𝑥⟩ ↔ 𝑥 = (2nd𝑦)))
2322bicomd 224 . . . 4 (𝑦 ∈ ({1o} × V) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨1o, 𝑥⟩))
2423ad2antll 725 . . 3 ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({1o} × V))) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨1o, 𝑥⟩))
251, 7, 8, 24f1o2d 7262 . 2 (⊤ → inr:V–1-1-onto→({1o} × V))
2625mptru 1529 1 inr:V–1-1-onto→({1o} × V)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1522  wtru 1523  wcel 2081  Vcvv 3437  {csn 4476  cop 4482   × cxp 5446  1-1-ontowf1o 6229  cfv 6230  ωcom 7441  1st c1st 7548  2nd c2nd 7549  1oc1o 7951  inrcinr 9180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3710  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-om 7442  df-1st 7550  df-2nd 7551  df-1o 7958  df-inr 9183
This theorem is referenced by:  inrresf  9196  inrresf1  9197  djuin  9198  djuun  9206
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