Theorem djulf1o 9904

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 9894	. . 3 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2		0ex 5307	. . . . . 6 ⊢ ∅ ∈ V
3	2	snid 4664	. . . . 5 ⊢ ∅ ∈ {∅}
4		opelxpi 5713	. . . . 5 ⊢ ((∅ ∈ {∅} ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
5	3, 4	mpan 689	. . . 4 ⊢ (𝑥 ∈ V → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
6	5	adantl 483	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
7		fvexd 6904	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({∅} × V)) → (2nd ‘𝑦) ∈ V)
8		1st2nd2 8011	. . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
9		xp1st 8004	. . . . . . . . . 10 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) ∈ {∅})
10		elsni 4645	. . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {∅} → (1st ‘𝑦) = ∅)
11	9, 10	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) = ∅)
12	11	opeq1d 4879	. . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨∅, (2nd ‘𝑦)⟩)
13	8, 12	eqtrd 2773	. . . . . . 7 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨∅, (2nd ‘𝑦)⟩)
14	13	eqeq2d 2744	. . . . . 6 ⊢ (𝑦 ∈ ({∅} × V) → (⟨∅, 𝑥⟩ = 𝑦 ↔ ⟨∅, 𝑥⟩ = ⟨∅, (2nd ‘𝑦)⟩))
15		eqcom 2740	. . . . . 6 ⊢ (⟨∅, 𝑥⟩ = 𝑦 ↔ 𝑦 = ⟨∅, 𝑥⟩)
16		eqid 2733	. . . . . . 7 ⊢ ∅ = ∅
17		vex 3479	. . . . . . . 8 ⊢ 𝑥 ∈ V
18	2, 17	opth 5476	. . . . . . 7 ⊢ (⟨∅, 𝑥⟩ = ⟨∅, (2nd ‘𝑦)⟩ ↔ (∅ = ∅ ∧ 𝑥 = (2nd ‘𝑦)))
19	16, 18	mpbiran 708	. . . . . 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 728	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({∅} × V))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨∅, 𝑥⟩))
23	1, 6, 7, 22	f1o2d 7657	. 2 ⊢ (⊤ → inl:V–1-1-onto→({∅} × V))
24	23	mptru 1549	1 ⊢ inl:V–1-1-onto→({∅} × V)

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542  ⊤wtru 1543   ∈ wcel 2107  Vcvv 3475  ∅c0 4322  {csn 4628  ⟨cop 4634   × cxp 5674  –1-1-onto→wf1o 6540  ‘cfv 6541  1^st c1st 7970  2^nd c2nd 7971  inlcinl 9891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-1st 7972  df-2nd 7973  df-inl 9894

This theorem is referenced by:  inlresf  9906  inlresf1  9907  djuin  9910  djuun  9918