Theorem djurf1o 9951

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 9941	. . 3 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2		1onn 8677	. . . . . 6 ⊢ 1o ∈ ω
3		snidg 4665	. . . . . 6 ⊢ (1o ∈ ω → 1o ∈ {1o})
4	2, 3	ax-mp 5	. . . . 5 ⊢ 1o ∈ {1o}
5		opelxpi 5726	. . . . 5 ⊢ ((1o ∈ {1o} ∧ 𝑥 ∈ V) → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
6	4, 5	mpan 690	. . . 4 ⊢ (𝑥 ∈ V → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
7	6	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ V) → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
8		fvexd 6922	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({1o} × V)) → (2nd ‘𝑦) ∈ V)
9		1st2nd2 8052	. . . . . . . 8 ⊢ (𝑦 ∈ ({1o} × V) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
10		xp1st 8045	. . . . . . . . . 10 ⊢ (𝑦 ∈ ({1o} × V) → (1st ‘𝑦) ∈ {1o})
11		elsni 4648	. . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {1o} → (1st ‘𝑦) = 1o)
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ ({1o} × V) → (1st ‘𝑦) = 1o)
13	12	opeq1d 4884	. . . . . . . 8 ⊢ (𝑦 ∈ ({1o} × V) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨1o, (2nd ‘𝑦)⟩)
14	9, 13	eqtrd 2775	. . . . . . 7 ⊢ (𝑦 ∈ ({1o} × V) → 𝑦 = ⟨1o, (2nd ‘𝑦)⟩)
15	14	eqeq2d 2746	. . . . . 6 ⊢ (𝑦 ∈ ({1o} × V) → (⟨1o, 𝑥⟩ = 𝑦 ↔ ⟨1o, 𝑥⟩ = ⟨1o, (2nd ‘𝑦)⟩))
16		eqcom 2742	. . . . . 6 ⊢ (⟨1o, 𝑥⟩ = 𝑦 ↔ 𝑦 = ⟨1o, 𝑥⟩)
17		eqid 2735	. . . . . . 7 ⊢ 1o = 1o
18		1oex 8515	. . . . . . . 8 ⊢ 1o ∈ V
19		vex 3482	. . . . . . . 8 ⊢ 𝑥 ∈ V
20	18, 19	opth 5487	. . . . . . 7 ⊢ (⟨1o, 𝑥⟩ = ⟨1o, (2nd ‘𝑦)⟩ ↔ (1o = 1o ∧ 𝑥 = (2nd ‘𝑦)))
21	17, 20	mpbiran 709	. . . . . 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 729	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({1o} × V))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨1o, 𝑥⟩))
25	1, 7, 8, 24	f1o2d 7687	. 2 ⊢ (⊤ → inr:V–1-1-onto→({1o} × V))
26	25	mptru 1544	1 ⊢ inr:V–1-1-onto→({1o} × V)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537  ⊤wtru 1538   ∈ wcel 2106  Vcvv 3478  {csn 4631  ⟨cop 4637   × cxp 5687  –1-1-onto→wf1o 6562  ‘cfv 6563  ωcom 7887  1^st c1st 8011  2^nd c2nd 8012  1_oc1o 8498  inrcinr 9938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-inr 9941

This theorem is referenced by:  inrresf  9954  inrresf1  9955  djuin  9956  djuun  9964