Theorem onsucrn 41954

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 486	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑎 = suc 𝑥) → 𝑎 = suc 𝑥)
2		onsuc 7794	. . . . . . . 8 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
3	2	adantr 482	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑎 = suc 𝑥) → suc 𝑥 ∈ On)
4	1, 3	eqeltrd 2834	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑎 = suc 𝑥) → 𝑎 ∈ On)
5	4	rexlimiva 3148	. . . . 5 ⊢ (∃𝑥 ∈ On 𝑎 = suc 𝑥 → 𝑎 ∈ On)
6	5	pm4.71ri 562	. . . 4 ⊢ (∃𝑥 ∈ On 𝑎 = suc 𝑥 ↔ (𝑎 ∈ On ∧ ∃𝑥 ∈ On 𝑎 = suc 𝑥))
7		suceq 6427	. . . . . . 7 ⊢ (𝑥 = 𝑏 → suc 𝑥 = suc 𝑏)
8	7	eqeq2d 2744	. . . . . 6 ⊢ (𝑥 = 𝑏 → (𝑎 = suc 𝑥 ↔ 𝑎 = suc 𝑏))
9	8	cbvrexvw 3236	. . . . 5 ⊢ (∃𝑥 ∈ On 𝑎 = suc 𝑥 ↔ ∃𝑏 ∈ On 𝑎 = suc 𝑏)
10	9	anbi2i 624	. . . 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 5952	. 2 ⊢ ran 𝐹 = {𝑎 ∣ ∃𝑥 ∈ On 𝑎 = suc 𝑥}
15		df-rab 3434	. 2 ⊢ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} = {𝑎 ∣ (𝑎 ∈ On ∧ ∃𝑏 ∈ On 𝑎 = suc 𝑏)}
16	12, 14, 15	3eqtr4i 2771	1 ⊢ ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}

Syntax hints:   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  ∃wrex 3071  {crab 3433   ↦ cmpt 5230  ran crn 5676  Oncon0 6361  suc csuc 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-cnv 5683  df-dm 5685  df-rn 5686  df-ord 6364  df-on 6365  df-suc 6367

This theorem is referenced by:  onsucf1o  41955