Theorem djulf1o 9870

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 9860	. . 3 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2		0ex 5257	. . . . . 6 ⊢ ∅ ∈ V
3	2	snid 4621	. . . . 5 ⊢ ∅ ∈ {∅}
4		opelxpi 5684	. . . . 5 ⊢ ((∅ ∈ {∅} ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
5	3, 4	mpan 700	. . . 4 ⊢ (𝑥 ∈ V → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
6	5	adantl 485	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V))
7		fvexd 6882	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({∅} × V)) → (2nd ‘𝑦) ∈ V)
8		1st2nd2 8009	. . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
9		xp1st 8002	. . . . . . . . . 10 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) ∈ {∅})
10		elsni 4599	. . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {∅} → (1st ‘𝑦) = ∅)
11	9, 10	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) = ∅)
12	11	opeq1d 4837	. . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨∅, (2nd ‘𝑦)⟩)
13	8, 12	eqtrd 2797	. . . . . . 7 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨∅, (2nd ‘𝑦)⟩)
14	13	eqeq2d 2773	. . . . . 6 ⊢ (𝑦 ∈ ({∅} × V) → (⟨∅, 𝑥⟩ = 𝑦 ↔ ⟨∅, 𝑥⟩ = ⟨∅, (2nd ‘𝑦)⟩))
15		eqcom 2769	. . . . . 6 ⊢ (⟨∅, 𝑥⟩ = 𝑦 ↔ 𝑦 = ⟨∅, 𝑥⟩)
16		eqid 2762	. . . . . . 7 ⊢ ∅ = ∅
17		vex 3458	. . . . . . . 8 ⊢ 𝑥 ∈ V
18	2, 17	opth 5444	. . . . . . 7 ⊢ (⟨∅, 𝑥⟩ = ⟨∅, (2nd ‘𝑦)⟩ ↔ (∅ = ∅ ∧ 𝑥 = (2nd ‘𝑦)))
19	16, 18	mpbiran 719	. . . . . 6 ⊢ (⟨∅, 𝑥⟩ = ⟨∅, (2nd ‘𝑦)⟩ ↔ 𝑥 = (2nd ‘𝑦))
20	14, 15, 19	3bitr3g 315	. . . . 5 ⊢ (𝑦 ∈ ({∅} × V) → (𝑦 = ⟨∅, 𝑥⟩ ↔ 𝑥 = (2nd ‘𝑦)))
21	20	bicomd 225	. . . 4 ⊢ (𝑦 ∈ ({∅} × V) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨∅, 𝑥⟩))
22	21	ad2antll 739	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({∅} × V))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨∅, 𝑥⟩))
23	1, 6, 7, 22	f1o2d 7650	. 2 ⊢ (⊤ → inl:V–1-1-onto→({∅} × V))
24	23	mptru 1567	1 ⊢ inl:V–1-1-onto→({∅} × V)

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1560  ⊤wtru 1561   ∈ wcel 2142  Vcvv 3454  ∅c0 4285  {csn 4582  ⟨cop 4588   × cxp 5645  –1-1-onto→wf1o 6520  ‘cfv 6521  1^st c1st 7968  2^nd c2nd 7969  inlcinl 9857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-1st 7970  df-2nd 7971  df-inl 9860

This theorem is referenced by:  inlresf  9872  inlresf1  9873  djuin  9876  djuun  9884