Theorem onsucrn 43233

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 484	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑎 = suc 𝑥) → 𝑎 = suc 𝑥)
2		onsuc 7847	. . . . . . . 8 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
3	2	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑎 = suc 𝑥) → suc 𝑥 ∈ On)
4	1, 3	eqeltrd 2844	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑎 = suc 𝑥) → 𝑎 ∈ On)
5	4	rexlimiva 3153	. . . . 5 ⊢ (∃𝑥 ∈ On 𝑎 = suc 𝑥 → 𝑎 ∈ On)
6	5	pm4.71ri 560	. . . 4 ⊢ (∃𝑥 ∈ On 𝑎 = suc 𝑥 ↔ (𝑎 ∈ On ∧ ∃𝑥 ∈ On 𝑎 = suc 𝑥))
7		suceq 6461	. . . . . . 7 ⊢ (𝑥 = 𝑏 → suc 𝑥 = suc 𝑏)
8	7	eqeq2d 2751	. . . . . 6 ⊢ (𝑥 = 𝑏 → (𝑎 = suc 𝑥 ↔ 𝑎 = suc 𝑏))
9	8	cbvrexvw 3244	. . . . 5 ⊢ (∃𝑥 ∈ On 𝑎 = suc 𝑥 ↔ ∃𝑏 ∈ On 𝑎 = suc 𝑏)
10	9	anbi2i 622	. . . 4 ⊢ ((𝑎 ∈ On ∧ ∃𝑥 ∈ On 𝑎 = suc 𝑥) ↔ (𝑎 ∈ On ∧ ∃𝑏 ∈ On 𝑎 = suc 𝑏))
11	6, 10	bitri 275	. . 3 ⊢ (∃𝑥 ∈ On 𝑎 = suc 𝑥 ↔ (𝑎 ∈ On ∧ ∃𝑏 ∈ On 𝑎 = suc 𝑏))
12	11	abbii 2812	. 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 5980	. 2 ⊢ ran 𝐹 = {𝑎 ∣ ∃𝑥 ∈ On 𝑎 = suc 𝑥}
15		df-rab 3444	. 2 ⊢ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} = {𝑎 ∣ (𝑎 ∈ On ∧ ∃𝑏 ∈ On 𝑎 = suc 𝑏)}
16	12, 14, 15	3eqtr4i 2778	1 ⊢ ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}

Syntax hints:   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∃wrex 3076  {crab 3443   ↦ cmpt 5249  ran crn 5701  Oncon0 6395  suc csuc 6397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-cnv 5708  df-dm 5710  df-rn 5711  df-ord 6398  df-on 6399  df-suc 6401

This theorem is referenced by:  onsucf1o  43234