Theorem djurf1o 9858

Description: The right injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)

Assertion

Ref	Expression
djurf1o	⊢ inr:V–1-1-onto→({1o} × V)

Proof of Theorem djurf1o

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-inr 9848	. . 3 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2		1onn 8591	. . . . . 6 ⊢ 1o ∈ ω
3		snidg 4625	. . . . . 6 ⊢ (1o ∈ ω → 1o ∈ {1o})
4	2, 3	ax-mp 5	. . . . 5 ⊢ 1o ∈ {1o}
5		opelxpi 5675	. . . . 5 ⊢ ((1o ∈ {1o} ∧ 𝑥 ∈ V) → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
6	4, 5	mpan 688	. . . 4 ⊢ (𝑥 ∈ V → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
7	6	adantl 482	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ V) → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
8		fvexd 6862	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({1o} × V)) → (2nd ‘𝑦) ∈ V)
9		1st2nd2 7965	. . . . . . . 8 ⊢ (𝑦 ∈ ({1o} × V) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
10		xp1st 7958	. . . . . . . . . 10 ⊢ (𝑦 ∈ ({1o} × V) → (1st ‘𝑦) ∈ {1o})
11		elsni 4608	. . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {1o} → (1st ‘𝑦) = 1o)
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ ({1o} × V) → (1st ‘𝑦) = 1o)
13	12	opeq1d 4841	. . . . . . . 8 ⊢ (𝑦 ∈ ({1o} × V) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨1o, (2nd ‘𝑦)⟩)
14	9, 13	eqtrd 2771	. . . . . . 7 ⊢ (𝑦 ∈ ({1o} × V) → 𝑦 = ⟨1o, (2nd ‘𝑦)⟩)
15	14	eqeq2d 2742	. . . . . 6 ⊢ (𝑦 ∈ ({1o} × V) → (⟨1o, 𝑥⟩ = 𝑦 ↔ ⟨1o, 𝑥⟩ = ⟨1o, (2nd ‘𝑦)⟩))
16		eqcom 2738	. . . . . 6 ⊢ (⟨1o, 𝑥⟩ = 𝑦 ↔ 𝑦 = ⟨1o, 𝑥⟩)
17		eqid 2731	. . . . . . 7 ⊢ 1o = 1o
18		1oex 8427	. . . . . . . 8 ⊢ 1o ∈ V
19		vex 3450	. . . . . . . 8 ⊢ 𝑥 ∈ V
20	18, 19	opth 5438	. . . . . . 7 ⊢ (⟨1o, 𝑥⟩ = ⟨1o, (2nd ‘𝑦)⟩ ↔ (1o = 1o ∧ 𝑥 = (2nd ‘𝑦)))
21	17, 20	mpbiran 707	. . . . . 6 ⊢ (⟨1o, 𝑥⟩ = ⟨1o, (2nd ‘𝑦)⟩ ↔ 𝑥 = (2nd ‘𝑦))
22	15, 16, 21	3bitr3g 312	. . . . 5 ⊢ (𝑦 ∈ ({1o} × V) → (𝑦 = ⟨1o, 𝑥⟩ ↔ 𝑥 = (2nd ‘𝑦)))
23	22	bicomd 222	. . . 4 ⊢ (𝑦 ∈ ({1o} × V) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨1o, 𝑥⟩))
24	23	ad2antll 727	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({1o} × V))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨1o, 𝑥⟩))
25	1, 7, 8, 24	f1o2d 7612	. 2 ⊢ (⊤ → inr:V–1-1-onto→({1o} × V))
26	25	mptru 1548	1 ⊢ inr:V–1-1-onto→({1o} × V)

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541  ⊤wtru 1542   ∈ wcel 2106  Vcvv 3446  {csn 4591  ⟨cop 4597   × cxp 5636  –1-1-onto→wf1o 6500  ‘cfv 6501  ωcom 7807  1^st c1st 7924  2^nd c2nd 7925  1_oc1o 8410  inrcinr 9845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-om 7808  df-1st 7926  df-2nd 7927  df-1o 8417  df-inr 9848

This theorem is referenced by:  inrresf  9861  inrresf1  9862  djuin  9863  djuun  9871