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Theorem onsucf1o 42477
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 = (π‘₯ ∈ On ↦ suc π‘₯)
Assertion
Ref Expression
onsucf1o 𝐹:On–1-1-ontoβ†’{π‘Ž ∈ On ∣ βˆƒπ‘ ∈ On π‘Ž = suc 𝑏}
Distinct variable groups:   𝐹,π‘Ž,𝑏   π‘₯,π‘Ž,𝑏
Allowed substitution hint:   𝐹(π‘₯)

Proof of Theorem onsucf1o
StepHypRef Expression
1 onsucf1o.f . . . 4 𝐹 = (π‘₯ ∈ On ↦ suc π‘₯)
21fin1a2lem2 10391 . . 3 𝐹:On–1-1β†’On
3 f1fn 6778 . . 3 (𝐹:On–1-1β†’On β†’ 𝐹 Fn On)
42, 3ax-mp 5 . 2 𝐹 Fn On
51onsucrn 42476 . 2 ran 𝐹 = {π‘Ž ∈ On ∣ βˆƒπ‘ ∈ On π‘Ž = suc 𝑏}
61fin1a2lem1 10390 . . . . . 6 (π‘Ž ∈ On β†’ (πΉβ€˜π‘Ž) = suc π‘Ž)
71fin1a2lem1 10390 . . . . . 6 (𝑏 ∈ On β†’ (πΉβ€˜π‘) = suc 𝑏)
86, 7eqeqan12d 2738 . . . . 5 ((π‘Ž ∈ On ∧ 𝑏 ∈ On) β†’ ((πΉβ€˜π‘Ž) = (πΉβ€˜π‘) ↔ suc π‘Ž = suc 𝑏))
9 suc11 6461 . . . . 5 ((π‘Ž ∈ On ∧ 𝑏 ∈ On) β†’ (suc π‘Ž = suc 𝑏 ↔ π‘Ž = 𝑏))
108, 9bitrd 279 . . . 4 ((π‘Ž ∈ On ∧ 𝑏 ∈ On) β†’ ((πΉβ€˜π‘Ž) = (πΉβ€˜π‘) ↔ π‘Ž = 𝑏))
1110biimpd 228 . . 3 ((π‘Ž ∈ On ∧ 𝑏 ∈ On) β†’ ((πΉβ€˜π‘Ž) = (πΉβ€˜π‘) β†’ π‘Ž = 𝑏))
1211rgen2 3189 . 2 βˆ€π‘Ž ∈ On βˆ€π‘ ∈ On ((πΉβ€˜π‘Ž) = (πΉβ€˜π‘) β†’ π‘Ž = 𝑏)
13 dff1o6 7265 . 2 (𝐹:On–1-1-ontoβ†’{π‘Ž ∈ On ∣ βˆƒπ‘ ∈ On π‘Ž = suc 𝑏} ↔ (𝐹 Fn On ∧ ran 𝐹 = {π‘Ž ∈ On ∣ βˆƒπ‘ ∈ On π‘Ž = suc 𝑏} ∧ βˆ€π‘Ž ∈ On βˆ€π‘ ∈ On ((πΉβ€˜π‘Ž) = (πΉβ€˜π‘) β†’ π‘Ž = 𝑏)))
144, 5, 12, 13mpbir3an 1338 1 𝐹:On–1-1-ontoβ†’{π‘Ž ∈ On ∣ βˆƒπ‘ ∈ On π‘Ž = suc 𝑏}
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  βˆƒwrex 3062  {crab 3424   ↦ cmpt 5221  ran crn 5667  Oncon0 6354  suc csuc 6356   Fn wfn 6528  β€“1-1β†’wf1 6530  β€“1-1-ontoβ†’wf1o 6532  β€˜cfv 6533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  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)
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