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Mirrors > Home > MPE Home > Th. List > cfidm | Structured version Visualization version GIF version |
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 9664 | . . . 4 ⊢ (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝐴 ∈ On → (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴)) |
3 | cfsmo 9681 | . . . 4 ⊢ (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (cf‘𝐴)𝑥 ⊆ (𝑓‘𝑦))) | |
4 | cfon 9665 | . . . . 5 ⊢ (cf‘𝐴) ∈ On | |
5 | cfcoflem 9682 | . . . . 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 687 | . . . 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 3981 | . 2 ⊢ (𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴)) |
9 | cf0 9661 | . . 3 ⊢ (cf‘∅) = ∅ | |
10 | cff 9658 | . . . . . . 7 ⊢ cf:On⟶On | |
11 | 10 | fdmi 6517 | . . . . . 6 ⊢ dom cf = On |
12 | 11 | eleq2i 2901 | . . . . 5 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On) |
13 | ndmfv 6693 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅) | |
14 | 12, 13 | sylnbir 332 | . . . 4 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅) |
15 | 14 | fveq2d 6667 | . . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘∅)) |
16 | 9, 15, 14 | 3eqtr4a 2879 | . 2 ⊢ (¬ 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴)) |
17 | 8, 16 | pm2.61i 183 | 1 ⊢ (cf‘(cf‘𝐴)) = (cf‘𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1079 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 ⊆ wss 3933 ∅c0 4288 dom cdm 5548 Oncon0 6184 ⟶wf 6344 ‘cfv 6348 Smo wsmo 7971 cfccf 9354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-smo 7972 df-recs 7997 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-card 9356 df-cf 9358 df-acn 9359 |
This theorem is referenced by: (None) |
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