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Theorem djurf1o 9328
 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 9318 . . 3 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2 1onn 8250 . . . . . 6 1o ∈ ω
3 snidg 4559 . . . . . 6 (1o ∈ ω → 1o ∈ {1o})
42, 3ax-mp 5 . . . . 5 1o ∈ {1o}
5 opelxpi 5556 . . . . 5 ((1o ∈ {1o} ∧ 𝑥 ∈ V) → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
64, 5mpan 689 . . . 4 (𝑥 ∈ V → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
76adantl 485 . . 3 ((⊤ ∧ 𝑥 ∈ V) → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
8 fvexd 6660 . . 3 ((⊤ ∧ 𝑦 ∈ ({1o} × V)) → (2nd𝑦) ∈ V)
9 1st2nd2 7712 . . . . . . . 8 (𝑦 ∈ ({1o} × V) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
10 xp1st 7705 . . . . . . . . . 10 (𝑦 ∈ ({1o} × V) → (1st𝑦) ∈ {1o})
11 elsni 4542 . . . . . . . . . 10 ((1st𝑦) ∈ {1o} → (1st𝑦) = 1o)
1210, 11syl 17 . . . . . . . . 9 (𝑦 ∈ ({1o} × V) → (1st𝑦) = 1o)
1312opeq1d 4771 . . . . . . . 8 (𝑦 ∈ ({1o} × V) → ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨1o, (2nd𝑦)⟩)
149, 13eqtrd 2833 . . . . . . 7 (𝑦 ∈ ({1o} × V) → 𝑦 = ⟨1o, (2nd𝑦)⟩)
1514eqeq2d 2809 . . . . . 6 (𝑦 ∈ ({1o} × V) → (⟨1o, 𝑥⟩ = 𝑦 ↔ ⟨1o, 𝑥⟩ = ⟨1o, (2nd𝑦)⟩))
16 eqcom 2805 . . . . . 6 (⟨1o, 𝑥⟩ = 𝑦𝑦 = ⟨1o, 𝑥⟩)
17 eqid 2798 . . . . . . 7 1o = 1o
18 1oex 8095 . . . . . . . 8 1o ∈ V
19 vex 3444 . . . . . . . 8 𝑥 ∈ V
2018, 19opth 5333 . . . . . . 7 (⟨1o, 𝑥⟩ = ⟨1o, (2nd𝑦)⟩ ↔ (1o = 1o𝑥 = (2nd𝑦)))
2117, 20mpbiran 708 . . . . . 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 728 . . 3 ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({1o} × V))) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨1o, 𝑥⟩))
251, 7, 8, 24f1o2d 7380 . 2 (⊤ → inr:V–1-1-onto→({1o} × V))
2625mptru 1545 1 inr:V–1-1-onto→({1o} × V)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538  ⊤wtru 1539   ∈ wcel 2111  Vcvv 3441  {csn 4525  ⟨cop 4531   × cxp 5517  –1-1-onto→wf1o 6323  ‘cfv 6324  ωcom 7562  1st c1st 7671  2nd c2nd 7672  1oc1o 8080  inrcinr 9315 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7563  df-1st 7673  df-2nd 7674  df-1o 8087  df-inr 9318 This theorem is referenced by:  inrresf  9331  inrresf1  9332  djuin  9333  djuun  9341
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