Theorem djurf1o 9839

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 9829	. . 3 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2		1onn 8580	. . . . . 6 ⊢ 1o ∈ ω
3		snidg 4619	. . . . . 6 ⊢ (1o ∈ ω → 1o ∈ {1o})
4	2, 3	ax-mp 5	. . . . 5 ⊢ 1o ∈ {1o}
5		opelxpi 5671	. . . . 5 ⊢ ((1o ∈ {1o} ∧ 𝑥 ∈ V) → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
6	4, 5	mpan 691	. . . 4 ⊢ (𝑥 ∈ V → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
7	6	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ V) → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
8		fvexd 6859	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({1o} × V)) → (2nd ‘𝑦) ∈ V)
9		1st2nd2 7984	. . . . . . . 8 ⊢ (𝑦 ∈ ({1o} × V) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
10		xp1st 7977	. . . . . . . . . 10 ⊢ (𝑦 ∈ ({1o} × V) → (1st ‘𝑦) ∈ {1o})
11		elsni 4599	. . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {1o} → (1st ‘𝑦) = 1o)
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ ({1o} × V) → (1st ‘𝑦) = 1o)
13	12	opeq1d 4837	. . . . . . . 8 ⊢ (𝑦 ∈ ({1o} × V) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨1o, (2nd ‘𝑦)⟩)
14	9, 13	eqtrd 2772	. . . . . . 7 ⊢ (𝑦 ∈ ({1o} × V) → 𝑦 = ⟨1o, (2nd ‘𝑦)⟩)
15	14	eqeq2d 2748	. . . . . 6 ⊢ (𝑦 ∈ ({1o} × V) → (⟨1o, 𝑥⟩ = 𝑦 ↔ ⟨1o, 𝑥⟩ = ⟨1o, (2nd ‘𝑦)⟩))
16		eqcom 2744	. . . . . 6 ⊢ (⟨1o, 𝑥⟩ = 𝑦 ↔ 𝑦 = ⟨1o, 𝑥⟩)
17		eqid 2737	. . . . . . 7 ⊢ 1o = 1o
18		1oex 8419	. . . . . . . 8 ⊢ 1o ∈ V
19		vex 3446	. . . . . . . 8 ⊢ 𝑥 ∈ V
20	18, 19	opth 5434	. . . . . . 7 ⊢ (⟨1o, 𝑥⟩ = ⟨1o, (2nd ‘𝑦)⟩ ↔ (1o = 1o ∧ 𝑥 = (2nd ‘𝑦)))
21	17, 20	mpbiran 710	. . . . . 6 ⊢ (⟨1o, 𝑥⟩ = ⟨1o, (2nd ‘𝑦)⟩ ↔ 𝑥 = (2nd ‘𝑦))
22	15, 16, 21	3bitr3g 313	. . . . 5 ⊢ (𝑦 ∈ ({1o} × V) → (𝑦 = ⟨1o, 𝑥⟩ ↔ 𝑥 = (2nd ‘𝑦)))
23	22	bicomd 223	. . . 4 ⊢ (𝑦 ∈ ({1o} × V) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨1o, 𝑥⟩))
24	23	ad2antll 730	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({1o} × V))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨1o, 𝑥⟩))
25	1, 7, 8, 24	f1o2d 7624	. 2 ⊢ (⊤ → inr:V–1-1-onto→({1o} × V))
26	25	mptru 1549	1 ⊢ inr:V–1-1-onto→({1o} × V)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114  Vcvv 3442  {csn 4582  ⟨cop 4588   × cxp 5632  –1-1-onto→wf1o 6501  ‘cfv 6502  ωcom 7820  1^st c1st 7943  2^nd c2nd 7944  1_oc1o 8402  inrcinr 9826

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-om 7821  df-1st 7945  df-2nd 7946  df-1o 8409  df-inr 9829

This theorem is referenced by:  inrresf  9842  inrresf1  9843  djuin  9844  djuun  9852