Theorem cfidm 10392

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 10370	. . . 4 ⊢ (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴)
2	1	a1i 11	. . 3 ⊢ (𝐴 ∈ On → (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴))
3		cfsmo 10388	. . . 4 ⊢ (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (cf‘𝐴)𝑥 ⊆ (𝑓‘𝑦)))
4		cfon 10371	. . . . 5 ⊢ (cf‘𝐴) ∈ On
5		cfcoflem 10389	. . . . 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 704	. . . 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 16	. . 3 ⊢ (𝐴 ∈ On → (cf‘𝐴) ⊆ (cf‘(cf‘𝐴)))
8	2, 7	eqssd 3930	. 2 ⊢ (𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴))
9		cf0 10367	. . 3 ⊢ (cf‘∅) = ∅
10		cff 10364	. . . . . . 7 ⊢ cf:On⟶On
11	10	fdmi 6751	. . . . . 6 ⊢ dom cf = On
12	11	eleq2i 2856	. . . . 5 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
13		ndmfv 6948	. . . . 5 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
14	12, 13	sylnbir 334	. . . 4 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅)
15	14	fveq2d 6920	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘∅))
16	9, 15, 14	3eqtr4a 2824	. 2 ⊢ (¬ 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴))
17	8, 16	pm2.61i 184	1 ⊢ (cf‘(cf‘𝐴)) = (cf‘𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900  ∅c0 4281  dom cdm 5658  Oncon0 6388  ⟶wf 6562  ‘cfv 6566  Smo wsmo 8391  cfccf 10055

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-smo 8392  df-recs 8417  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-card 10057  df-cf 10059  df-acn 10060

This theorem is used by: (None)