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Theorem cfidm 8980
 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 8959 . . . 4 (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴)
21a1i 11 . . 3 (𝐴 ∈ On → (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴))
3 cfsmo 8976 . . . 4 (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦 ∈ (cf‘𝐴)𝑥 ⊆ (𝑓𝑦)))
4 cfon 8960 . . . . 5 (cf‘𝐴) ∈ On
5 cfcoflem 8977 . . . . 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 703 . . . 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 3585 . 2 (𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴))
9 cf0 8956 . . 3 (cf‘∅) = ∅
10 cff 8953 . . . . . . 7 cf:On⟶On
1110fdmi 5965 . . . . . 6 dom cf = On
1211eleq2i 2680 . . . . 5 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
13 ndmfv 6128 . . . . 5 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
1412, 13sylnbir 320 . . . 4 𝐴 ∈ On → (cf‘𝐴) = ∅)
1514fveq2d 6107 . . 3 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘∅))
169, 15, 143eqtr4a 2670 . 2 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴))
178, 16pm2.61i 175 1 (cf‘(cf‘𝐴)) = (cf‘𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874  dom cdm 5038  Oncon0 5640  ⟶wf 5800  ‘cfv 5804  Smo wsmo 7329  cfccf 8646 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-smo 7330  df-recs 7355  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-card 8648  df-cf 8650  df-acn 8651 This theorem is referenced by: (None)
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