Theorem djurf1o 9932

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 9922	. . 3 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2		1onn 8657	. . . . . 6 ⊢ 1o ∈ ω
3		snidg 4641	. . . . . 6 ⊢ (1o ∈ ω → 1o ∈ {1o})
4	2, 3	ax-mp 5	. . . . 5 ⊢ 1o ∈ {1o}
5		opelxpi 5696	. . . . 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 6896	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({1o} × V)) → (2nd ‘𝑦) ∈ V)
9		1st2nd2 8032	. . . . . . . 8 ⊢ (𝑦 ∈ ({1o} × V) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
10		xp1st 8025	. . . . . . . . . 10 ⊢ (𝑦 ∈ ({1o} × V) → (1st ‘𝑦) ∈ {1o})
11		elsni 4623	. . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {1o} → (1st ‘𝑦) = 1o)
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ ({1o} × V) → (1st ‘𝑦) = 1o)
13	12	opeq1d 4860	. . . . . . . 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 2747	. . . . . 6 ⊢ (𝑦 ∈ ({1o} × V) → (⟨1o, 𝑥⟩ = 𝑦 ↔ ⟨1o, 𝑥⟩ = ⟨1o, (2nd ‘𝑦)⟩))
16		eqcom 2743	. . . . . 6 ⊢ (⟨1o, 𝑥⟩ = 𝑦 ↔ 𝑦 = ⟨1o, 𝑥⟩)
17		eqid 2736	. . . . . . 7 ⊢ 1o = 1o
18		1oex 8495	. . . . . . . 8 ⊢ 1o ∈ V
19		vex 3468	. . . . . . . 8 ⊢ 𝑥 ∈ V
20	18, 19	opth 5456	. . . . . . 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 7666	. 2 ⊢ (⊤ → inr:V–1-1-onto→({1o} × V))
26	25	mptru 1547	1 ⊢ inr:V–1-1-onto→({1o} × V)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109  Vcvv 3464  {csn 4606  ⟨cop 4612   × cxp 5657  –1-1-onto→wf1o 6535  ‘cfv 6536  ωcom 7866  1^st c1st 7991  2^nd c2nd 7992  1_oc1o 8478  inrcinr 9919

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7867  df-1st 7993  df-2nd 7994  df-1o 8485  df-inr 9922

This theorem is referenced by:  inrresf  9935  inrresf1  9936  djuin  9937  djuun  9945