Theorem cfidm 10183

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.

Step	Hyp	Ref	Expression
1		cfle 10162	. . . 4 ⊢ (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴)
2	1	a1i 11	. . 3 ⊢ (𝐴 ∈ On → (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴))
3		cfsmo 10179	. . . 4 ⊢ (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (cf‘𝐴)𝑥 ⊆ (𝑓‘𝑦)))
4		cfon 10163	. . . . 5 ⊢ (cf‘𝐴) ∈ On
5		cfcoflem 10180	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (cf‘𝐴) ∈ On) → (∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (cf‘𝐴)𝑥 ⊆ (𝑓‘𝑦)) → (cf‘𝐴) ⊆ (cf‘(cf‘𝐴))))
6	4, 5	mpan2 691	. . . 4 ⊢ (𝐴 ∈ On → (∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (cf‘𝐴)𝑥 ⊆ (𝑓‘𝑦)) → (cf‘𝐴) ⊆ (cf‘(cf‘𝐴))))
7	3, 6	mpd 15	. . 3 ⊢ (𝐴 ∈ On → (cf‘𝐴) ⊆ (cf‘(cf‘𝐴)))
8	2, 7	eqssd 3949	. 2 ⊢ (𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴))
9		cf0 10159	. . 3 ⊢ (cf‘∅) = ∅
10		cff 10156	. . . . . . 7 ⊢ cf:On⟶On
11	10	fdmi 6671	. . . . . 6 ⊢ dom cf = On
12	11	eleq2i 2826	. . . . 5 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
13		ndmfv 6864	. . . . 5 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
14	12, 13	sylnbir 331	. . . 4 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅)
15	14	fveq2d 6836	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘∅))
16	9, 15, 14	3eqtr4a 2795	. 2 ⊢ (¬ 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴))
17	8, 16	pm2.61i 182	1 ⊢ (cf‘(cf‘𝐴)) = (cf‘𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058   ⊆ wss 3899  ∅c0 4283  dom cdm 5622  Oncon0 6315  ⟶wf 6486  ‘cfv 6490  Smo wsmo 8275  cfccf 9847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-smo 8276  df-recs 8301  df-er 8633  df-map 8763  df-en 8882  df-dom 8883  df-sdom 8884  df-card 9849  df-cf 9851  df-acn 9852

This theorem is referenced by: (None)