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Theorem cfidm 9298
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 9277 . . . 4 (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴)
21a1i 11 . . 3 (𝐴 ∈ On → (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴))
3 cfsmo 9294 . . . 4 (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦 ∈ (cf‘𝐴)𝑥 ⊆ (𝑓𝑦)))
4 cfon 9278 . . . . 5 (cf‘𝐴) ∈ On
5 cfcoflem 9295 . . . . 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 663 . . . 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 3767 . 2 (𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴))
9 cf0 9274 . . 3 (cf‘∅) = ∅
10 cff 9271 . . . . . . 7 cf:On⟶On
1110fdmi 6192 . . . . . 6 dom cf = On
1211eleq2i 2841 . . . . 5 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
13 ndmfv 6359 . . . . 5 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
1412, 13sylnbir 320 . . . 4 𝐴 ∈ On → (cf‘𝐴) = ∅)
1514fveq2d 6336 . . 3 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘∅))
169, 15, 143eqtr4a 2830 . 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 1070   = wceq 1630  wex 1851  wcel 2144  wral 3060  wrex 3061  wss 3721  c0 4061  dom cdm 5249  Oncon0 5866  wf 6027  cfv 6031  Smo wsmo 7594  cfccf 8962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-smo 7595  df-recs 7620  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-card 8964  df-cf 8966  df-acn 8967
This theorem is referenced by: (None)
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