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Theorem djurf1o 9529
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 9519 . . 3 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2 1onn 8367 . . . . . 6 1o ∈ ω
3 snidg 4575 . . . . . 6 (1o ∈ ω → 1o ∈ {1o})
42, 3ax-mp 5 . . . . 5 1o ∈ {1o}
5 opelxpi 5588 . . . . 5 ((1o ∈ {1o} ∧ 𝑥 ∈ V) → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
64, 5mpan 690 . . . 4 (𝑥 ∈ V → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
76adantl 485 . . 3 ((⊤ ∧ 𝑥 ∈ V) → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
8 fvexd 6732 . . 3 ((⊤ ∧ 𝑦 ∈ ({1o} × V)) → (2nd𝑦) ∈ V)
9 1st2nd2 7800 . . . . . . . 8 (𝑦 ∈ ({1o} × V) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
10 xp1st 7793 . . . . . . . . . 10 (𝑦 ∈ ({1o} × V) → (1st𝑦) ∈ {1o})
11 elsni 4558 . . . . . . . . . 10 ((1st𝑦) ∈ {1o} → (1st𝑦) = 1o)
1210, 11syl 17 . . . . . . . . 9 (𝑦 ∈ ({1o} × V) → (1st𝑦) = 1o)
1312opeq1d 4790 . . . . . . . 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 8215 . . . . . . . 8 1o ∈ V
19 vex 3412 . . . . . . . 8 𝑥 ∈ V
2018, 19opth 5360 . . . . . . 7 (⟨1o, 𝑥⟩ = ⟨1o, (2nd𝑦)⟩ ↔ (1o = 1o𝑥 = (2nd𝑦)))
2117, 20mpbiran 709 . . . . . 6 (⟨1o, 𝑥⟩ = ⟨1o, (2nd𝑦)⟩ ↔ 𝑥 = (2nd𝑦))
2215, 16, 213bitr3g 316 . . . . 5 (𝑦 ∈ ({1o} × V) → (𝑦 = ⟨1o, 𝑥⟩ ↔ 𝑥 = (2nd𝑦)))
2322bicomd 226 . . . 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 7459 . 2 (⊤ → inr:V–1-1-onto→({1o} × V))
2625mptru 1550 1 inr:V–1-1-onto→({1o} × V)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1543  wtru 1544  wcel 2110  Vcvv 3408  {csn 4541  cop 4547   × cxp 5549  1-1-ontowf1o 6379  cfv 6380  ωcom 7644  1st c1st 7759  2nd c2nd 7760  1oc1o 8195  inrcinr 9516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-om 7645  df-1st 7761  df-2nd 7762  df-1o 8202  df-inr 9519
This theorem is referenced by:  inrresf  9532  inrresf1  9533  djuin  9534  djuun  9542
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