Theorem onsucf1o 41955

Description: The successor operation is a bijective function between the ordinals and the class of succesor 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 10392	. . 3 ⊢ 𝐹:On–1-1→On
3		f1fn 6785	. . 3 ⊢ (𝐹:On–1-1→On → 𝐹 Fn On)
4	2, 3	ax-mp 5	. 2 ⊢ 𝐹 Fn On
5	1	onsucrn 41954	. 2 ⊢ ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}
6	1	fin1a2lem1 10391	. . . . . 6 ⊢ (𝑎 ∈ On → (𝐹‘𝑎) = suc 𝑎)
7	1	fin1a2lem1 10391	. . . . . 6 ⊢ (𝑏 ∈ On → (𝐹‘𝑏) = suc 𝑏)
8	6, 7	eqeqan12d 2747	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ suc 𝑎 = suc 𝑏))
9		suc11 6468	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (suc 𝑎 = suc 𝑏 ↔ 𝑎 = 𝑏))
10	8, 9	bitrd 279	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ 𝑎 = 𝑏))
11	10	biimpd 228	. . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
12	11	rgen2 3198	. 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)
13		dff1o6 7268	. 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 1342	1 ⊢ 𝐹:On–1-1-onto→{𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  {crab 3433   ↦ cmpt 5230  ran crn 5676  Oncon0 6361  suc csuc 6363   Fn wfn 6535  –1-1→wf1 6537  –1-1-onto→wf1o 6539  ‘cfv 6540

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-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548

This theorem is referenced by: (None)