Theorem onsucf1o 43262

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 10439	. . 3 ⊢ 𝐹:On–1-1→On
3		f1fn 6806	. . 3 ⊢ (𝐹:On–1-1→On → 𝐹 Fn On)
4	2, 3	ax-mp 5	. 2 ⊢ 𝐹 Fn On
5	1	onsucrn 43261	. 2 ⊢ ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}
6	1	fin1a2lem1 10438	. . . . . 6 ⊢ (𝑎 ∈ On → (𝐹‘𝑎) = suc 𝑎)
7	1	fin1a2lem1 10438	. . . . . 6 ⊢ (𝑏 ∈ On → (𝐹‘𝑏) = suc 𝑏)
8	6, 7	eqeqan12d 2749	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ suc 𝑎 = suc 𝑏))
9		suc11 6493	. . . . 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 3197	. 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)
13		dff1o6 7295	. 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 1340	1 ⊢ 𝐹:On–1-1-onto→{𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  {crab 3433   ↦ cmpt 5231  ran crn 5690  Oncon0 6386  suc csuc 6388   Fn wfn 6558  –1-1→wf1 6560  –1-1-onto→wf1o 6562  ‘cfv 6563

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-res 5701  df-ima 5702  df-ord 6389  df-on 6390  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

This theorem is referenced by: (None)