Theorem onsucrn 44113

Description: The successor operation is surjective onto its range, the class of successor ordinals. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.)

Hypothesis

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

Assertion

Ref	Expression
onsucrn	⊢ ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}

Distinct variable group:   𝑎,𝑏,𝑥

Allowed substitution hints:   𝐹(𝑥, 𝑎, 𝑏)

Proof of Theorem onsucrn

Step	Hyp	Ref	Expression
1		simpr 490	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑎 = suc 𝑥) → 𝑎 = suc 𝑥)
2		onsuc 7810	. . . . . . . 8 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
3	2	adantr 486	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑎 = suc 𝑥) → suc 𝑥 ∈ On)
4	1, 3	eqeltrd 2860	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑎 = suc 𝑥) → 𝑎 ∈ On)
5	4	rexlimiva 3155	. . . . 5 ⊢ (∃𝑥 ∈ On 𝑎 = suc 𝑥 → 𝑎 ∈ On)
6	5	pm4.71ri 570	. . . 4 ⊢ (∃𝑥 ∈ On 𝑎 = suc 𝑥 ↔ (𝑎 ∈ On ∧ ∃𝑥 ∈ On 𝑎 = suc 𝑥))
7		suceq 6426	. . . . . . 7 ⊢ (𝑥 = 𝑏 → suc 𝑥 = suc 𝑏)
8	7	eqeq2d 2771	. . . . . 6 ⊢ (𝑥 = 𝑏 → (𝑎 = suc 𝑥 ↔ 𝑎 = suc 𝑏))
9	8	cbvrexvw 3241	. . . . 5 ⊢ (∃𝑥 ∈ On 𝑎 = suc 𝑥 ↔ ∃𝑏 ∈ On 𝑎 = suc 𝑏)
10	9	anbi2i 635	. . . 4 ⊢ ((𝑎 ∈ On ∧ ∃𝑥 ∈ On 𝑎 = suc 𝑥) ↔ (𝑎 ∈ On ∧ ∃𝑏 ∈ On 𝑎 = suc 𝑏))
11	6, 10	bitri 278	. . 3 ⊢ (∃𝑥 ∈ On 𝑎 = suc 𝑥 ↔ (𝑎 ∈ On ∧ ∃𝑏 ∈ On 𝑎 = suc 𝑏))
12	11	abbii 2827	. 2 ⊢ {𝑎 ∣ ∃𝑥 ∈ On 𝑎 = suc 𝑥} = {𝑎 ∣ (𝑎 ∈ On ∧ ∃𝑏 ∈ On 𝑎 = suc 𝑏)}
13		onsucrn.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ On ↦ suc 𝑥)
14	13	rnmpt 5941	. 2 ⊢ ran 𝐹 = {𝑎 ∣ ∃𝑥 ∈ On 𝑎 = suc 𝑥}
15		df-rab 3413	. 2 ⊢ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} = {𝑎 ∣ (𝑎 ∈ On ∧ ∃𝑏 ∈ On 𝑎 = suc 𝑏)}
16	12, 14, 15	3eqtr4i 2793	1 ⊢ ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738  ∃wrex 3086  {crab 3412   ↦ cmpt 5186  ran crn 5656  Oncon0 6357  suc csuc 6359

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-cnv 5663  df-dm 5665  df-rn 5666  df-ord 6360  df-on 6361  df-suc 6363

This theorem is used by:  onsucf1o  44114