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.

Step	Hyp	Ref	Expression
1		df-inl 9943	. . 3 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2		0ex 5306	. . . . . 6 ⊢ ∅ ∈ V
3	2	snid 4661	. . . . 5 ⊢ ∅ ∈ {∅}
4		opelxpi 5721	. . . . 5 ⊢ ((∅ ∈ {∅} ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
5	3, 4	mpan 690	. . . 4 ⊢ (𝑥 ∈ V → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
6	5	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
7		fvexd 6920	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({∅} × V)) → (2nd ‘𝑦) ∈ V)
8		1st2nd2 8054	. . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
9		xp1st 8047	. . . . . . . . . 10 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) ∈ {∅})
10		elsni 4642	. . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {∅} → (1st ‘𝑦) = ∅)
11	9, 10	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) = ∅)
12	11	opeq1d 4878	. . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨∅, (2nd ‘𝑦)⟩)
13	8, 12	eqtrd 2776	. . . . . . 7 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨∅, (2nd ‘𝑦)⟩)
14	13	eqeq2d 2747	. . . . . 6 ⊢ (𝑦 ∈ ({∅} × V) → (⟨∅, 𝑥⟩ = 𝑦 ↔ ⟨∅, 𝑥⟩ = ⟨∅, (2nd ‘𝑦)⟩))
15		eqcom 2743	. . . . . 6 ⊢ (⟨∅, 𝑥⟩ = 𝑦 ↔ 𝑦 = ⟨∅, 𝑥⟩)
16		eqid 2736	. . . . . . 7 ⊢ ∅ = ∅
17		vex 3483	. . . . . . . 8 ⊢ 𝑥 ∈ V
18	2, 17	opth 5480	. . . . . . 7 ⊢ (⟨∅, 𝑥⟩ = ⟨∅, (2nd ‘𝑦)⟩ ↔ (∅ = ∅ ∧ 𝑥 = (2nd ‘𝑦)))
19	16, 18	mpbiran 709	. . . . . 6 ⊢ (⟨∅, 𝑥⟩ = ⟨∅, (2nd ‘𝑦)⟩ ↔ 𝑥 = (2nd ‘𝑦))
20	14, 15, 19	3bitr3g 313	. . . . 5 ⊢ (𝑦 ∈ ({∅} × V) → (𝑦 = ⟨∅, 𝑥⟩ ↔ 𝑥 = (2nd ‘𝑦)))
21	20	bicomd 223	. . . 4 ⊢ (𝑦 ∈ ({∅} × V) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨∅, 𝑥⟩))
22	21	ad2antll 729	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({∅} × V))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨∅, 𝑥⟩))
23	1, 6, 7, 22	f1o2d 7688	. 2 ⊢ (⊤ → inl:V–1-1-onto→({∅} × V))
24	23	mptru 1546	1 ⊢ inl:V–1-1-onto→({∅} × V)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1539  ⊤wtru 1540   ∈ wcel 2107  Vcvv 3479  ∅c0 4332  {csn 4625  ⟨cop 4631   × cxp 5682  –1-1-onto→wf1o 6559  ‘cfv 6560  1^st c1st 8013  2^nd 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