MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  djulf1o Structured version   Visualization version   GIF version

Theorem djulf1o 9943
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 9933 . . 3 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2 0ex 5311 . . . . . 6 ∅ ∈ V
32snid 4669 . . . . 5 ∅ ∈ {∅}
4 opelxpi 5719 . . . . 5 ((∅ ∈ {∅} ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
53, 4mpan 688 . . . 4 (𝑥 ∈ V → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
65adantl 480 . . 3 ((⊤ ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
7 fvexd 6917 . . 3 ((⊤ ∧ 𝑦 ∈ ({∅} × V)) → (2nd𝑦) ∈ V)
8 1st2nd2 8038 . . . . . . . 8 (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
9 xp1st 8031 . . . . . . . . . 10 (𝑦 ∈ ({∅} × V) → (1st𝑦) ∈ {∅})
10 elsni 4649 . . . . . . . . . 10 ((1st𝑦) ∈ {∅} → (1st𝑦) = ∅)
119, 10syl 17 . . . . . . . . 9 (𝑦 ∈ ({∅} × V) → (1st𝑦) = ∅)
1211opeq1d 4884 . . . . . . . 8 (𝑦 ∈ ({∅} × V) → ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨∅, (2nd𝑦)⟩)
138, 12eqtrd 2768 . . . . . . 7 (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨∅, (2nd𝑦)⟩)
1413eqeq2d 2739 . . . . . 6 (𝑦 ∈ ({∅} × V) → (⟨∅, 𝑥⟩ = 𝑦 ↔ ⟨∅, 𝑥⟩ = ⟨∅, (2nd𝑦)⟩))
15 eqcom 2735 . . . . . 6 (⟨∅, 𝑥⟩ = 𝑦𝑦 = ⟨∅, 𝑥⟩)
16 eqid 2728 . . . . . . 7 ∅ = ∅
17 vex 3477 . . . . . . . 8 𝑥 ∈ V
182, 17opth 5482 . . . . . . 7 (⟨∅, 𝑥⟩ = ⟨∅, (2nd𝑦)⟩ ↔ (∅ = ∅ ∧ 𝑥 = (2nd𝑦)))
1916, 18mpbiran 707 . . . . . 6 (⟨∅, 𝑥⟩ = ⟨∅, (2nd𝑦)⟩ ↔ 𝑥 = (2nd𝑦))
2014, 15, 193bitr3g 312 . . . . 5 (𝑦 ∈ ({∅} × V) → (𝑦 = ⟨∅, 𝑥⟩ ↔ 𝑥 = (2nd𝑦)))
2120bicomd 222 . . . 4 (𝑦 ∈ ({∅} × V) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨∅, 𝑥⟩))
2221ad2antll 727 . . 3 ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({∅} × V))) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨∅, 𝑥⟩))
231, 6, 7, 22f1o2d 7681 . 2 (⊤ → inl:V–1-1-onto→({∅} × V))
2423mptru 1540 1 inl:V–1-1-onto→({∅} × V)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  wtru 1534  wcel 2098  Vcvv 3473  c0 4326  {csn 4632  cop 4638   × cxp 5680  1-1-ontowf1o 6552  cfv 6553  1st c1st 7997  2nd c2nd 7998  inlcinl 9930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-1st 7999  df-2nd 8000  df-inl 9933
This theorem is referenced by:  inlresf  9945  inlresf1  9946  djuin  9949  djuun  9957
  Copyright terms: Public domain W3C validator