Theorem djurf1o 9910

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 9900	. . 3 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2		1onn 8641	. . . . . 6 ⊢ 1o ∈ ω
3		snidg 4661	. . . . . 6 ⊢ (1o ∈ ω → 1o ∈ {1o})
4	2, 3	ax-mp 5	. . . . 5 ⊢ 1o ∈ {1o}
5		opelxpi 5712	. . . . 5 ⊢ ((1o ∈ {1o} ∧ 𝑥 ∈ V) → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
6	4, 5	mpan 686	. . . 4 ⊢ (𝑥 ∈ V → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
7	6	adantl 480	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ V) → ⟨1o, 𝑥⟩ ∈ ({1o} × V))
8		fvexd 6905	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({1o} × V)) → (2nd ‘𝑦) ∈ V)
9		1st2nd2 8016	. . . . . . . 8 ⊢ (𝑦 ∈ ({1o} × V) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
10		xp1st 8009	. . . . . . . . . 10 ⊢ (𝑦 ∈ ({1o} × V) → (1st ‘𝑦) ∈ {1o})
11		elsni 4644	. . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {1o} → (1st ‘𝑦) = 1o)
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ ({1o} × V) → (1st ‘𝑦) = 1o)
13	12	opeq1d 4878	. . . . . . . 8 ⊢ (𝑦 ∈ ({1o} × V) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨1o, (2nd ‘𝑦)⟩)
14	9, 13	eqtrd 2770	. . . . . . 7 ⊢ (𝑦 ∈ ({1o} × V) → 𝑦 = ⟨1o, (2nd ‘𝑦)⟩)
15	14	eqeq2d 2741	. . . . . 6 ⊢ (𝑦 ∈ ({1o} × V) → (⟨1o, 𝑥⟩ = 𝑦 ↔ ⟨1o, 𝑥⟩ = ⟨1o, (2nd ‘𝑦)⟩))
16		eqcom 2737	. . . . . 6 ⊢ (⟨1o, 𝑥⟩ = 𝑦 ↔ 𝑦 = ⟨1o, 𝑥⟩)
17		eqid 2730	. . . . . . 7 ⊢ 1o = 1o
18		1oex 8478	. . . . . . . 8 ⊢ 1o ∈ V
19		vex 3476	. . . . . . . 8 ⊢ 𝑥 ∈ V
20	18, 19	opth 5475	. . . . . . 7 ⊢ (⟨1o, 𝑥⟩ = ⟨1o, (2nd ‘𝑦)⟩ ↔ (1o = 1o ∧ 𝑥 = (2nd ‘𝑦)))
21	17, 20	mpbiran 705	. . . . . 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 725	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({1o} × V))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨1o, 𝑥⟩))
25	1, 7, 8, 24	f1o2d 7662	. 2 ⊢ (⊤ → inr:V–1-1-onto→({1o} × V))
26	25	mptru 1546	1 ⊢ inr:V–1-1-onto→({1o} × V)

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1539  ⊤wtru 1540   ∈ wcel 2104  Vcvv 3472  {csn 4627  ⟨cop 4633   × cxp 5673  –1-1-onto→wf1o 6541  ‘cfv 6542  ωcom 7857  1^st c1st 7975  2^nd c2nd 7976  1_oc1o 8461  inrcinr 9897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7858  df-1st 7977  df-2nd 7978  df-1o 8468  df-inr 9900

This theorem is referenced by:  inrresf  9913  inrresf1  9914  djuin  9915  djuun  9923