Theorem djulf1o 9897

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 9887	. . 3 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2		0ex 5272	. . . . . 6 ⊢ ∅ ∈ V
3	2	snid 4633	. . . . 5 ⊢ ∅ ∈ {∅}
4		opelxpi 5699	. . . . 5 ⊢ ((∅ ∈ {∅} ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
5	3, 4	mpan 702	. . . 4 ⊢ (𝑥 ∈ V → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
6	5	adantl 486	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
7		fvexd 6897	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({∅} × V)) → (2nd ‘𝑦) ∈ V)
8		1st2nd2 8024	. . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
9		xp1st 8017	. . . . . . . . . 10 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) ∈ {∅})
10		elsni 4611	. . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {∅} → (1st ‘𝑦) = ∅)
11	9, 10	syl 18	. . . . . . . . 9 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) = ∅)
12	11	opeq1d 4848	. . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨∅, (2nd ‘𝑦)⟩)
13	8, 12	eqtrd 2804	. . . . . . 7 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨∅, (2nd ‘𝑦)⟩)
14	13	eqeq2d 2780	. . . . . 6 ⊢ (𝑦 ∈ ({∅} × V) → (⟨∅, 𝑥⟩ = 𝑦 ↔ ⟨∅, 𝑥⟩ = ⟨∅, (2nd ‘𝑦)⟩))
15		eqcom 2776	. . . . . 6 ⊢ (⟨∅, 𝑥⟩ = 𝑦 ↔ 𝑦 = ⟨∅, 𝑥⟩)
16		eqid 2769	. . . . . . 7 ⊢ ∅ = ∅
17		vex 3467	. . . . . . . 8 ⊢ 𝑥 ∈ V
18	2, 17	opth 5459	. . . . . . 7 ⊢ (⟨∅, 𝑥⟩ = ⟨∅, (2nd ‘𝑦)⟩ ↔ (∅ = ∅ ∧ 𝑥 = (2nd ‘𝑦)))
19	16, 18	mpbiran 721	. . . . . 6 ⊢ (⟨∅, 𝑥⟩ = ⟨∅, (2nd ‘𝑦)⟩ ↔ 𝑥 = (2nd ‘𝑦))
20	14, 15, 19	3bitr3g 316	. . . . 5 ⊢ (𝑦 ∈ ({∅} × V) → (𝑦 = ⟨∅, 𝑥⟩ ↔ 𝑥 = (2nd ‘𝑦)))
21	20	bicomd 226	. . . 4 ⊢ (𝑦 ∈ ({∅} × V) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨∅, 𝑥⟩))
22	21	ad2antll 741	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({∅} × V))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨∅, 𝑥⟩))
23	1, 6, 7, 22	f1o2d 7665	. 2 ⊢ (⊤ → inl:V–1-1-onto→({∅} × V))
24	23	mptru 1574	1 ⊢ inl:V–1-1-onto→({∅} × V)

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567  ⊤wtru 1568   ∈ wcel 2149  Vcvv 3463  ∅c0 4294  {csn 4594  ⟨cop 4600   × cxp 5660  –1-1-onto→wf1o 6536  ‘cfv 6537  1^st c1st 7983  2^nd c2nd 7984  inlcinl 9884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-1st 7985  df-2nd 7986  df-inl 9887

This theorem is referenced by:  inlresf  9899  inlresf1  9900  djuin  9903  djuun  9911