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| Description: The cofinality function is idempotent. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 15-Sep-2013.) | 
| Ref | Expression | 
|---|---|
| cfidm | ⊢ (cf‘(cf‘𝐴)) = (cf‘𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cfle 10294 | . . . 4 ⊢ (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴) | |
| 2 | 1 | a1i 11 | . . 3 ⊢ (𝐴 ∈ On → (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴)) | 
| 3 | cfsmo 10311 | . . . 4 ⊢ (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (cf‘𝐴)𝑥 ⊆ (𝑓‘𝑦))) | |
| 4 | cfon 10295 | . . . . 5 ⊢ (cf‘𝐴) ∈ On | |
| 5 | cfcoflem 10312 | . . . . 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 4001 | . 2 ⊢ (𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴)) | 
| 9 | cf0 10291 | . . 3 ⊢ (cf‘∅) = ∅ | |
| 10 | cff 10288 | . . . . . . 7 ⊢ cf:On⟶On | |
| 11 | 10 | fdmi 6747 | . . . . . 6 ⊢ dom cf = On | 
| 12 | 11 | eleq2i 2833 | . . . . 5 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On) | 
| 13 | ndmfv 6941 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅) | |
| 14 | 12, 13 | sylnbir 331 | . . . 4 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅) | 
| 15 | 14 | fveq2d 6910 | . . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘∅)) | 
| 16 | 9, 15, 14 | 3eqtr4a 2803 | . 2 ⊢ (¬ 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴)) | 
| 17 | 8, 16 | pm2.61i 182 | 1 ⊢ (cf‘(cf‘𝐴)) = (cf‘𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1087 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 ∅c0 4333 dom cdm 5685 Oncon0 6384 ⟶wf 6557 ‘cfv 6561 Smo wsmo 8385 cfccf 9977 | 
| 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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-smo 8386 df-recs 8411 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-card 9979 df-cf 9981 df-acn 9982 | 
| This theorem is referenced by: (None) | 
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