Theorem onsucrn 43262

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 7810	. . . . . . . 8 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
3	2	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑎 = suc 𝑥) → suc 𝑥 ∈ On)
4	1, 3	eqeltrd 2835	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑎 = suc 𝑥) → 𝑎 ∈ On)
5	4	rexlimiva 3134	. . . . 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 6424	. . . . . . 7 ⊢ (𝑥 = 𝑏 → suc 𝑥 = suc 𝑏)
8	7	eqeq2d 2747	. . . . . 6 ⊢ (𝑥 = 𝑏 → (𝑎 = suc 𝑥 ↔ 𝑎 = suc 𝑏))
9	8	cbvrexvw 3225	. . . . 5 ⊢ (∃𝑥 ∈ On 𝑎 = suc 𝑥 ↔ ∃𝑏 ∈ On 𝑎 = suc 𝑏)
10	9	anbi2i 623	. . . 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 2803	. 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 5942	. 2 ⊢ ran 𝐹 = {𝑎 ∣ ∃𝑥 ∈ On 𝑎 = suc 𝑥}
15		df-rab 3421	. 2 ⊢ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} = {𝑎 ∣ (𝑎 ∈ On ∧ ∃𝑏 ∈ On 𝑎 = suc 𝑏)}
16	12, 14, 15	3eqtr4i 2769	1 ⊢ ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2714  ∃wrex 3061  {crab 3420   ↦ cmpt 5206  ran crn 5660  Oncon0 6357  suc csuc 6359

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 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-cnv 5667  df-dm 5669  df-rn 5670  df-ord 6360  df-on 6361  df-suc 6363

This theorem is referenced by:  onsucf1o  43263