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Theorem djulf1o 8942
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 8933 . . 3 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2 0ex 4925 . . . . . 6 ∅ ∈ V
32snid 4348 . . . . 5 ∅ ∈ {∅}
4 opelxpi 5287 . . . . 5 ((∅ ∈ {∅} ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
53, 4mpan 670 . . . 4 (𝑥 ∈ V → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
65adantl 467 . . 3 ((⊤ ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
7 xp2nd 7352 . . . 4 (𝑦 ∈ ({∅} × V) → (2nd𝑦) ∈ V)
87adantl 467 . . 3 ((⊤ ∧ 𝑦 ∈ ({∅} × V)) → (2nd𝑦) ∈ V)
9 1st2nd2 7358 . . . . . . . 8 (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
10 xp1st 7351 . . . . . . . . . 10 (𝑦 ∈ ({∅} × V) → (1st𝑦) ∈ {∅})
11 elsni 4334 . . . . . . . . . 10 ((1st𝑦) ∈ {∅} → (1st𝑦) = ∅)
1210, 11syl 17 . . . . . . . . 9 (𝑦 ∈ ({∅} × V) → (1st𝑦) = ∅)
1312opeq1d 4546 . . . . . . . 8 (𝑦 ∈ ({∅} × V) → ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨∅, (2nd𝑦)⟩)
149, 13eqtrd 2805 . . . . . . 7 (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨∅, (2nd𝑦)⟩)
1514eqeq2d 2781 . . . . . 6 (𝑦 ∈ ({∅} × V) → (⟨∅, 𝑥⟩ = 𝑦 ↔ ⟨∅, 𝑥⟩ = ⟨∅, (2nd𝑦)⟩))
16 eqcom 2778 . . . . . 6 (⟨∅, 𝑥⟩ = 𝑦𝑦 = ⟨∅, 𝑥⟩)
17 eqid 2771 . . . . . . 7 ∅ = ∅
18 vex 3354 . . . . . . . 8 𝑥 ∈ V
192, 18opth 5073 . . . . . . 7 (⟨∅, 𝑥⟩ = ⟨∅, (2nd𝑦)⟩ ↔ (∅ = ∅ ∧ 𝑥 = (2nd𝑦)))
2017, 19mpbiran 688 . . . . . 6 (⟨∅, 𝑥⟩ = ⟨∅, (2nd𝑦)⟩ ↔ 𝑥 = (2nd𝑦))
2115, 16, 203bitr3g 302 . . . . 5 (𝑦 ∈ ({∅} × V) → (𝑦 = ⟨∅, 𝑥⟩ ↔ 𝑥 = (2nd𝑦)))
2221bicomd 213 . . . 4 (𝑦 ∈ ({∅} × V) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨∅, 𝑥⟩))
2322ad2antll 708 . . 3 ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({∅} × V))) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨∅, 𝑥⟩))
241, 6, 8, 23f1o2d 7038 . 2 (⊤ → inl:V–1-1-onto→({∅} × V))
2524trud 1641 1 inl:V–1-1-onto→({∅} × V)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1631  wtru 1632  wcel 2145  Vcvv 3351  c0 4063  {csn 4317  cop 4323   × cxp 5248  1-1-ontowf1o 6029  cfv 6030  1st c1st 7317  2nd c2nd 7318  inlcinl 8930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-1st 7319  df-2nd 7320  df-inl 8933
This theorem is referenced by:  inlresf  8944  inlresf1  8945  djuin  8948  djuun  8956
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