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Theorem cfidm 9701
 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 9680 . . . 4 (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴)
21a1i 11 . . 3 (𝐴 ∈ On → (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴))
3 cfsmo 9697 . . . 4 (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦 ∈ (cf‘𝐴)𝑥 ⊆ (𝑓𝑦)))
4 cfon 9681 . . . . 5 (cf‘𝐴) ∈ On
5 cfcoflem 9698 . . . . 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 690 . . . 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 3933 . 2 (𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴))
9 cf0 9677 . . 3 (cf‘∅) = ∅
10 cff 9674 . . . . . . 7 cf:On⟶On
1110fdmi 6503 . . . . . 6 dom cf = On
1211eleq2i 2881 . . . . 5 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
13 ndmfv 6682 . . . . 5 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
1412, 13sylnbir 334 . . . 4 𝐴 ∈ On → (cf‘𝐴) = ∅)
1514fveq2d 6656 . . 3 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘∅))
169, 15, 143eqtr4a 2859 . 2 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴))
178, 16pm2.61i 185 1 (cf‘(cf‘𝐴)) = (cf‘𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ⊆ wss 3882  ∅c0 4245  dom cdm 5522  Oncon0 6164  ⟶wf 6325  ‘cfv 6329  Smo wsmo 7980  cfccf 9365 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7451 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3722  df-csb 3830  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-pss 3901  df-nul 4246  df-if 4428  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5441  df-so 5442  df-fr 5481  df-se 5482  df-we 5483  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-pred 6121  df-ord 6167  df-on 6168  df-suc 6170  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-isom 6338  df-riota 7100  df-ov 7145  df-oprab 7146  df-mpo 7147  df-1st 7681  df-2nd 7682  df-wrecs 7945  df-smo 7981  df-recs 8006  df-er 8287  df-map 8406  df-en 8508  df-dom 8509  df-sdom 8510  df-card 9367  df-cf 9369  df-acn 9370 This theorem is referenced by: (None)
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