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Theorem cfidm 10315
Description: The cofinality function is idempotent. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
Assertion
Ref Expression
cfidm (cf‘(cf‘𝐴)) = (cf‘𝐴)

Proof of Theorem cfidm
Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cfle 10294 . . . 4 (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴)
21a1i 11 . . 3 (𝐴 ∈ On → (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴))
3 cfsmo 10311 . . . 4 (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦 ∈ (cf‘𝐴)𝑥 ⊆ (𝑓𝑦)))
4 cfon 10295 . . . . 5 (cf‘𝐴) ∈ On
5 cfcoflem 10312 . . . . 5 ((𝐴 ∈ On ∧ (cf‘𝐴) ∈ On) → (∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦 ∈ (cf‘𝐴)𝑥 ⊆ (𝑓𝑦)) → (cf‘𝐴) ⊆ (cf‘(cf‘𝐴))))
64, 5mpan2 691 . . . 4 (𝐴 ∈ On → (∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦 ∈ (cf‘𝐴)𝑥 ⊆ (𝑓𝑦)) → (cf‘𝐴) ⊆ (cf‘(cf‘𝐴))))
73, 6mpd 15 . . 3 (𝐴 ∈ On → (cf‘𝐴) ⊆ (cf‘(cf‘𝐴)))
82, 7eqssd 4001 . 2 (𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴))
9 cf0 10291 . . 3 (cf‘∅) = ∅
10 cff 10288 . . . . . . 7 cf:On⟶On
1110fdmi 6747 . . . . . 6 dom cf = On
1211eleq2i 2833 . . . . 5 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
13 ndmfv 6941 . . . . 5 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
1412, 13sylnbir 331 . . . 4 𝐴 ∈ On → (cf‘𝐴) = ∅)
1514fveq2d 6910 . . 3 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘∅))
169, 15, 143eqtr4a 2803 . 2 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴))
178, 16pm2.61i 182 1 (cf‘(cf‘𝐴)) = (cf‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wral 3061  wrex 3070  wss 3951  c0 4333  dom cdm 5685  Oncon0 6384  wf 6557  cfv 6561  Smo wsmo 8385  cfccf 9977
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-smo 8386  df-recs 8411  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-card 9979  df-cf 9981  df-acn 9982
This theorem is referenced by: (None)
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