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Theorem djulf1o 9953
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 9943 . . 3 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2 0ex 5306 . . . . . 6 ∅ ∈ V
32snid 4661 . . . . 5 ∅ ∈ {∅}
4 opelxpi 5721 . . . . 5 ((∅ ∈ {∅} ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
53, 4mpan 690 . . . 4 (𝑥 ∈ V → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
65adantl 481 . . 3 ((⊤ ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
7 fvexd 6920 . . 3 ((⊤ ∧ 𝑦 ∈ ({∅} × V)) → (2nd𝑦) ∈ V)
8 1st2nd2 8054 . . . . . . . 8 (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
9 xp1st 8047 . . . . . . . . . 10 (𝑦 ∈ ({∅} × V) → (1st𝑦) ∈ {∅})
10 elsni 4642 . . . . . . . . . 10 ((1st𝑦) ∈ {∅} → (1st𝑦) = ∅)
119, 10syl 17 . . . . . . . . 9 (𝑦 ∈ ({∅} × V) → (1st𝑦) = ∅)
1211opeq1d 4878 . . . . . . . 8 (𝑦 ∈ ({∅} × V) → ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨∅, (2nd𝑦)⟩)
138, 12eqtrd 2776 . . . . . . 7 (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨∅, (2nd𝑦)⟩)
1413eqeq2d 2747 . . . . . 6 (𝑦 ∈ ({∅} × V) → (⟨∅, 𝑥⟩ = 𝑦 ↔ ⟨∅, 𝑥⟩ = ⟨∅, (2nd𝑦)⟩))
15 eqcom 2743 . . . . . 6 (⟨∅, 𝑥⟩ = 𝑦𝑦 = ⟨∅, 𝑥⟩)
16 eqid 2736 . . . . . . 7 ∅ = ∅
17 vex 3483 . . . . . . . 8 𝑥 ∈ V
182, 17opth 5480 . . . . . . 7 (⟨∅, 𝑥⟩ = ⟨∅, (2nd𝑦)⟩ ↔ (∅ = ∅ ∧ 𝑥 = (2nd𝑦)))
1916, 18mpbiran 709 . . . . . 6 (⟨∅, 𝑥⟩ = ⟨∅, (2nd𝑦)⟩ ↔ 𝑥 = (2nd𝑦))
2014, 15, 193bitr3g 313 . . . . 5 (𝑦 ∈ ({∅} × V) → (𝑦 = ⟨∅, 𝑥⟩ ↔ 𝑥 = (2nd𝑦)))
2120bicomd 223 . . . 4 (𝑦 ∈ ({∅} × V) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨∅, 𝑥⟩))
2221ad2antll 729 . . 3 ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({∅} × V))) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨∅, 𝑥⟩))
231, 6, 7, 22f1o2d 7688 . 2 (⊤ → inl:V–1-1-onto→({∅} × V))
2423mptru 1546 1 inl:V–1-1-onto→({∅} × V)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1539  wtru 1540  wcel 2107  Vcvv 3479  c0 4332  {csn 4625  cop 4631   × cxp 5682  1-1-ontowf1o 6559  cfv 6560  1st c1st 8013  2nd c2nd 8014  inlcinl 9940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-1st 8015  df-2nd 8016  df-inl 9943
This theorem is referenced by:  inlresf  9955  inlresf1  9956  djuin  9959  djuun  9967
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