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Theorem cfidm 9135
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 9114 . . . 4 (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴)
21a1i 11 . . 3 (𝐴 ∈ On → (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴))
3 cfsmo 9131 . . . 4 (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦 ∈ (cf‘𝐴)𝑥 ⊆ (𝑓𝑦)))
4 cfon 9115 . . . . 5 (cf‘𝐴) ∈ On
5 cfcoflem 9132 . . . . 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 707 . . . 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 3653 . 2 (𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴))
9 cf0 9111 . . 3 (cf‘∅) = ∅
10 cff 9108 . . . . . . 7 cf:On⟶On
1110fdmi 6090 . . . . . 6 dom cf = On
1211eleq2i 2722 . . . . 5 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
13 ndmfv 6256 . . . . 5 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
1412, 13sylnbir 320 . . . 4 𝐴 ∈ On → (cf‘𝐴) = ∅)
1514fveq2d 6233 . . 3 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘∅))
169, 15, 143eqtr4a 2711 . 2 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴))
178, 16pm2.61i 176 1 (cf‘(cf‘𝐴)) = (cf‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wral 2941  wrex 2942  wss 3607  c0 3948  dom cdm 5143  Oncon0 5761  wf 5922  cfv 5926  Smo wsmo 7487  cfccf 8801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-smo 7488  df-recs 7513  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-card 8803  df-cf 8805  df-acn 8806
This theorem is referenced by: (None)
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