Theorem onsucf1o 43245

Description: The successor operation is a bijective function between the ordinals and the class of successor ordinals. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.)

Hypothesis

Ref	Expression
onsucf1o.f	⊢ 𝐹 = (𝑥 ∈ On ↦ suc 𝑥)

Assertion

Ref	Expression
onsucf1o	⊢ 𝐹:On–1-1-onto→{𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}

Distinct variable groups:   𝐹,𝑎,𝑏   𝑥,𝑎,𝑏

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem onsucf1o

Step	Hyp	Ref	Expression
1		onsucf1o.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ On ↦ suc 𝑥)
2	1	fin1a2lem2 10295	. . 3 ⊢ 𝐹:On–1-1→On
3		f1fn 6721	. . 3 ⊢ (𝐹:On–1-1→On → 𝐹 Fn On)
4	2, 3	ax-mp 5	. 2 ⊢ 𝐹 Fn On
5	1	onsucrn 43244	. 2 ⊢ ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}
6	1	fin1a2lem1 10294	. . . . . 6 ⊢ (𝑎 ∈ On → (𝐹‘𝑎) = suc 𝑎)
7	1	fin1a2lem1 10294	. . . . . 6 ⊢ (𝑏 ∈ On → (𝐹‘𝑏) = suc 𝑏)
8	6, 7	eqeqan12d 2743	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ suc 𝑎 = suc 𝑏))
9		suc11 6416	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (suc 𝑎 = suc 𝑏 ↔ 𝑎 = 𝑏))
10	8, 9	bitrd 279	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ 𝑎 = 𝑏))
11	10	biimpd 229	. . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
12	11	rgen2 3169	. 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)
13		dff1o6 7212	. 2 ⊢ (𝐹:On–1-1-onto→{𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} ↔ (𝐹 Fn On ∧ ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} ∧ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
14	4, 5, 12, 13	mpbir3an 1342	1 ⊢ 𝐹:On–1-1-onto→{𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3394   ↦ cmpt 5173  ran crn 5620  Oncon0 6307  suc csuc 6309   Fn wfn 6477  –1-1→wf1 6479  –1-1-onto→wf1o 6481  ‘cfv 6482

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 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490

This theorem is referenced by: (None)