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Theorem djulf1o 9194
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 9184 . . 3 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2 0ex 5109 . . . . . 6 ∅ ∈ V
32snid 4512 . . . . 5 ∅ ∈ {∅}
4 opelxpi 5487 . . . . 5 ((∅ ∈ {∅} ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
53, 4mpan 686 . . . 4 (𝑥 ∈ V → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
65adantl 482 . . 3 ((⊤ ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
7 fvexd 6560 . . 3 ((⊤ ∧ 𝑦 ∈ ({∅} × V)) → (2nd𝑦) ∈ V)
8 1st2nd2 7591 . . . . . . . 8 (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
9 xp1st 7584 . . . . . . . . . 10 (𝑦 ∈ ({∅} × V) → (1st𝑦) ∈ {∅})
10 elsni 4495 . . . . . . . . . 10 ((1st𝑦) ∈ {∅} → (1st𝑦) = ∅)
119, 10syl 17 . . . . . . . . 9 (𝑦 ∈ ({∅} × V) → (1st𝑦) = ∅)
1211opeq1d 4722 . . . . . . . 8 (𝑦 ∈ ({∅} × V) → ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨∅, (2nd𝑦)⟩)
138, 12eqtrd 2833 . . . . . . 7 (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨∅, (2nd𝑦)⟩)
1413eqeq2d 2807 . . . . . 6 (𝑦 ∈ ({∅} × V) → (⟨∅, 𝑥⟩ = 𝑦 ↔ ⟨∅, 𝑥⟩ = ⟨∅, (2nd𝑦)⟩))
15 eqcom 2804 . . . . . 6 (⟨∅, 𝑥⟩ = 𝑦𝑦 = ⟨∅, 𝑥⟩)
16 eqid 2797 . . . . . . 7 ∅ = ∅
17 vex 3443 . . . . . . . 8 𝑥 ∈ V
182, 17opth 5267 . . . . . . 7 (⟨∅, 𝑥⟩ = ⟨∅, (2nd𝑦)⟩ ↔ (∅ = ∅ ∧ 𝑥 = (2nd𝑦)))
1916, 18mpbiran 705 . . . . . 6 (⟨∅, 𝑥⟩ = ⟨∅, (2nd𝑦)⟩ ↔ 𝑥 = (2nd𝑦))
2014, 15, 193bitr3g 314 . . . . 5 (𝑦 ∈ ({∅} × V) → (𝑦 = ⟨∅, 𝑥⟩ ↔ 𝑥 = (2nd𝑦)))
2120bicomd 224 . . . 4 (𝑦 ∈ ({∅} × V) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨∅, 𝑥⟩))
2221ad2antll 725 . . 3 ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({∅} × V))) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨∅, 𝑥⟩))
231, 6, 7, 22f1o2d 7264 . 2 (⊤ → inl:V–1-1-onto→({∅} × V))
2423mptru 1532 1 inl:V–1-1-onto→({∅} × V)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1525  wtru 1526  wcel 2083  Vcvv 3440  c0 4217  {csn 4478  cop 4484   × cxp 5448  1-1-ontowf1o 6231  cfv 6232  1st c1st 7550  2nd c2nd 7551  inlcinl 9181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-1st 7552  df-2nd 7553  df-inl 9184
This theorem is referenced by:  inlresf  9196  inlresf1  9197  djuin  9200  djuun  9208
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