Theorem onsucf1o 44019

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 10380	. . 3 ⊢ 𝐹:On–1-1→On
3		f1fn 6775	. . 3 ⊢ (𝐹:On–1-1→On → 𝐹 Fn On)
4	2, 3	ax-mp 5	. 2 ⊢ 𝐹 Fn On
5	1	onsucrn 44018	. 2 ⊢ ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}
6	1	fin1a2lem1 10379	. . . . . 6 ⊢ (𝑎 ∈ On → (𝐹‘𝑎) = suc 𝑎)
7	1	fin1a2lem1 10379	. . . . . 6 ⊢ (𝑏 ∈ On → (𝐹‘𝑏) = suc 𝑏)
8	6, 7	eqeqan12d 2777	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ suc 𝑎 = suc 𝑏))
9		suc11 6470	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (suc 𝑎 = suc 𝑏 ↔ 𝑎 = 𝑏))
10	8, 9	bitrd 282	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ 𝑎 = 𝑏))
11	10	biimpd 232	. . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
12	11	rgen2 3205	. 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)
13		dff1o6 7273	. 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 1360	1 ⊢ 𝐹:On–1-1-onto→{𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  {crab 3416   ↦ cmpt 5192  ran crn 5662  Oncon0 6360  suc csuc 6362   Fn wfn 6531  –1-1→wf1 6533  –1-1-onto→wf1o 6535  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544

This theorem is referenced by: (None)