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Theorem djulf1o 9343
 Description: The left injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)
Assertion
Ref Expression
djulf1o inl:V–1-1-onto→({∅} × V)

Proof of Theorem djulf1o
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-inl 9333 . . 3 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2 0ex 5179 . . . . . 6 ∅ ∈ V
32snid 4564 . . . . 5 ∅ ∈ {∅}
4 opelxpi 5560 . . . . 5 ((∅ ∈ {∅} ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
53, 4mpan 689 . . . 4 (𝑥 ∈ V → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
65adantl 485 . . 3 ((⊤ ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
7 fvexd 6670 . . 3 ((⊤ ∧ 𝑦 ∈ ({∅} × V)) → (2nd𝑦) ∈ V)
8 1st2nd2 7723 . . . . . . . 8 (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
9 xp1st 7716 . . . . . . . . . 10 (𝑦 ∈ ({∅} × V) → (1st𝑦) ∈ {∅})
10 elsni 4545 . . . . . . . . . 10 ((1st𝑦) ∈ {∅} → (1st𝑦) = ∅)
119, 10syl 17 . . . . . . . . 9 (𝑦 ∈ ({∅} × V) → (1st𝑦) = ∅)
1211opeq1d 4775 . . . . . . . 8 (𝑦 ∈ ({∅} × V) → ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨∅, (2nd𝑦)⟩)
138, 12eqtrd 2833 . . . . . . 7 (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨∅, (2nd𝑦)⟩)
1413eqeq2d 2809 . . . . . 6 (𝑦 ∈ ({∅} × V) → (⟨∅, 𝑥⟩ = 𝑦 ↔ ⟨∅, 𝑥⟩ = ⟨∅, (2nd𝑦)⟩))
15 eqcom 2805 . . . . . 6 (⟨∅, 𝑥⟩ = 𝑦𝑦 = ⟨∅, 𝑥⟩)
16 eqid 2798 . . . . . . 7 ∅ = ∅
17 vex 3445 . . . . . . . 8 𝑥 ∈ V
182, 17opth 5337 . . . . . . 7 (⟨∅, 𝑥⟩ = ⟨∅, (2nd𝑦)⟩ ↔ (∅ = ∅ ∧ 𝑥 = (2nd𝑦)))
1916, 18mpbiran 708 . . . . . 6 (⟨∅, 𝑥⟩ = ⟨∅, (2nd𝑦)⟩ ↔ 𝑥 = (2nd𝑦))
2014, 15, 193bitr3g 316 . . . . 5 (𝑦 ∈ ({∅} × V) → (𝑦 = ⟨∅, 𝑥⟩ ↔ 𝑥 = (2nd𝑦)))
2120bicomd 226 . . . 4 (𝑦 ∈ ({∅} × V) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨∅, 𝑥⟩))
2221ad2antll 728 . . 3 ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({∅} × V))) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨∅, 𝑥⟩))
231, 6, 7, 22f1o2d 7390 . 2 (⊤ → inl:V–1-1-onto→({∅} × V))
2423mptru 1545 1 inl:V–1-1-onto→({∅} × V)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538  ⊤wtru 1539   ∈ wcel 2111  Vcvv 3442  ∅c0 4246  {csn 4528  ⟨cop 4534   × cxp 5521  –1-1-onto→wf1o 6331  ‘cfv 6332  1st c1st 7682  2nd c2nd 7683  inlcinl 9330 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 5171  ax-nul 5178  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  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 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-1st 7684  df-2nd 7685  df-inl 9333 This theorem is referenced by:  inlresf  9345  inlresf1  9346  djuin  9349  djuun  9357
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