Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  onsucf1o Structured version   Visualization version   GIF version

Theorem onsucf1o 41857
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
StepHypRef Expression
1 onsucf1o.f . . . 4 𝐹 = (𝑥 ∈ On ↦ suc 𝑥)
21fin1a2lem2 10380 . . 3 𝐹:On–1-1→On
3 f1fn 6776 . . 3 (𝐹:On–1-1→On → 𝐹 Fn On)
42, 3ax-mp 5 . 2 𝐹 Fn On
51onsucrn 41856 . 2 ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}
61fin1a2lem1 10379 . . . . . 6 (𝑎 ∈ On → (𝐹𝑎) = suc 𝑎)
71fin1a2lem1 10379 . . . . . 6 (𝑏 ∈ On → (𝐹𝑏) = suc 𝑏)
86, 7eqeqan12d 2746 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝐹𝑎) = (𝐹𝑏) ↔ suc 𝑎 = suc 𝑏))
9 suc11 6461 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (suc 𝑎 = suc 𝑏𝑎 = 𝑏))
108, 9bitrd 278 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝐹𝑎) = (𝐹𝑏) ↔ 𝑎 = 𝑏))
1110biimpd 228 . . 3 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
1211rgen2 3197 . 2 𝑎 ∈ On ∀𝑏 ∈ On ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)
13 dff1o6 7258 . 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 ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
144, 5, 12, 13mpbir3an 1341 1 𝐹:On–1-1-onto→{𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3061  wrex 3070  {crab 3432  cmpt 5225  ran crn 5671  Oncon0 6354  suc csuc 6356   Fn wfn 6528  1-1wf1 6530  1-1-ontowf1o 6532  cfv 6533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-ord 6357  df-on 6358  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator