Theorem cfidm 10289

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 10268	. . . 4 ⊢ (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴)
2	1	a1i 11	. . 3 ⊢ (𝐴 ∈ On → (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴))
3		cfsmo 10285	. . . 4 ⊢ (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (cf‘𝐴)𝑥 ⊆ (𝑓‘𝑦)))
4		cfon 10269	. . . . 5 ⊢ (cf‘𝐴) ∈ On
5		cfcoflem 10286	. . . . 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 3976	. 2 ⊢ (𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴))
9		cf0 10265	. . 3 ⊢ (cf‘∅) = ∅
10		cff 10262	. . . . . . 7 ⊢ cf:On⟶On
11	10	fdmi 6717	. . . . . 6 ⊢ dom cf = On
12	11	eleq2i 2826	. . . . 5 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
13		ndmfv 6911	. . . . 5 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
14	12, 13	sylnbir 331	. . . 4 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅)
15	14	fveq2d 6880	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘∅))
16	9, 15, 14	3eqtr4a 2796	. 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 1540  ∃wex 1779   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060   ⊆ wss 3926  ∅c0 4308  dom cdm 5654  Oncon0 6352  ⟶wf 6527  ‘cfv 6531  Smo wsmo 8359  cfccf 9951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-smo 8360  df-recs 8385  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-card 9953  df-cf 9955  df-acn 9956

This theorem is referenced by: (None)