Theorem djulf1o 9906

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 9896	. . 3 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2		0ex 5300	. . . . . 6 ⊢ ∅ ∈ V
3	2	snid 4659	. . . . 5 ⊢ ∅ ∈ {∅}
4		opelxpi 5706	. . . . 5 ⊢ ((∅ ∈ {∅} ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
5	3, 4	mpan 687	. . . 4 ⊢ (𝑥 ∈ V → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
6	5	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
7		fvexd 6899	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({∅} × V)) → (2nd ‘𝑦) ∈ V)
8		1st2nd2 8010	. . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
9		xp1st 8003	. . . . . . . . . 10 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) ∈ {∅})
10		elsni 4640	. . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {∅} → (1st ‘𝑦) = ∅)
11	9, 10	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) = ∅)
12	11	opeq1d 4874	. . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨∅, (2nd ‘𝑦)⟩)
13	8, 12	eqtrd 2766	. . . . . . 7 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨∅, (2nd ‘𝑦)⟩)
14	13	eqeq2d 2737	. . . . . 6 ⊢ (𝑦 ∈ ({∅} × V) → (⟨∅, 𝑥⟩ = 𝑦 ↔ ⟨∅, 𝑥⟩ = ⟨∅, (2nd ‘𝑦)⟩))
15		eqcom 2733	. . . . . 6 ⊢ (⟨∅, 𝑥⟩ = 𝑦 ↔ 𝑦 = ⟨∅, 𝑥⟩)
16		eqid 2726	. . . . . . 7 ⊢ ∅ = ∅
17		vex 3472	. . . . . . . 8 ⊢ 𝑥 ∈ V
18	2, 17	opth 5469	. . . . . . 7 ⊢ (⟨∅, 𝑥⟩ = ⟨∅, (2nd ‘𝑦)⟩ ↔ (∅ = ∅ ∧ 𝑥 = (2nd ‘𝑦)))
19	16, 18	mpbiran 706	. . . . . 6 ⊢ (⟨∅, 𝑥⟩ = ⟨∅, (2nd ‘𝑦)⟩ ↔ 𝑥 = (2nd ‘𝑦))
20	14, 15, 19	3bitr3g 313	. . . . 5 ⊢ (𝑦 ∈ ({∅} × V) → (𝑦 = ⟨∅, 𝑥⟩ ↔ 𝑥 = (2nd ‘𝑦)))
21	20	bicomd 222	. . . 4 ⊢ (𝑦 ∈ ({∅} × V) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨∅, 𝑥⟩))
22	21	ad2antll 726	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({∅} × V))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨∅, 𝑥⟩))
23	1, 6, 7, 22	f1o2d 7656	. 2 ⊢ (⊤ → inl:V–1-1-onto→({∅} × V))
24	23	mptru 1540	1 ⊢ inl:V–1-1-onto→({∅} × V)

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533  ⊤wtru 1534   ∈ wcel 2098  Vcvv 3468  ∅c0 4317  {csn 4623  ⟨cop 4629   × cxp 5667  –1-1-onto→wf1o 6535  ‘cfv 6536  1^st c1st 7969  2^nd c2nd 7970  inlcinl 9893

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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-1st 7971  df-2nd 7972  df-inl 9896

This theorem is referenced by:  inlresf  9908  inlresf1  9909  djuin  9912  djuun  9920