Theorem cfidm 10346

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 10324	. . . 4 ⊢ (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴)
2	1	a1i 11	. . 3 ⊢ (𝐴 ∈ On → (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴))
3		cfsmo 10342	. . . 4 ⊢ (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (cf‘𝐴)𝑥 ⊆ (𝑓‘𝑦)))
4		cfon 10325	. . . . 5 ⊢ (cf‘𝐴) ∈ On
5		cfcoflem 10343	. . . . 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 3948	. 2 ⊢ (𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴))
9		cf0 10321	. . 3 ⊢ (cf‘∅) = ∅
10		cff 10318	. . . . . . 7 ⊢ cf:On⟶On
11	10	fdmi 6719	. . . . . 6 ⊢ dom cf = On
12	11	eleq2i 2853	. . . . 5 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
13		ndmfv 6915	. . . . 5 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
14	12, 13	sylnbir 334	. . . 4 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅)
15	14	fveq2d 6887	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘∅))
16	9, 15, 14	3eqtr4a 2822	. 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 2145  ∀wral 3077  ∃wrex 3087   ⊆ wss 3899  ∅c0 4279  dom cdm 5651  Oncon0 6361  ⟶wf 6533  ‘cfv 6537  Smo wsmo 8346  cfccf 10011

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-smo 8347  df-recs 8372  df-er 8710  df-map 8842  df-en 8967  df-dom 8968  df-sdom 8969  df-card 10013  df-cf 10015  df-acn 10016

This theorem is used by: (None)