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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 9678 | . . . 4 ⊢ (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝐴 ∈ On → (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴)) |
3 | cfsmo 9695 | . . . 4 ⊢ (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (cf‘𝐴)𝑥 ⊆ (𝑓‘𝑦))) | |
4 | cfon 9679 | . . . . 5 ⊢ (cf‘𝐴) ∈ On | |
5 | cfcoflem 9696 | . . . . 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 689 | . . . 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 3986 | . 2 ⊢ (𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴)) |
9 | cf0 9675 | . . 3 ⊢ (cf‘∅) = ∅ | |
10 | cff 9672 | . . . . . . 7 ⊢ cf:On⟶On | |
11 | 10 | fdmi 6526 | . . . . . 6 ⊢ dom cf = On |
12 | 11 | eleq2i 2906 | . . . . 5 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On) |
13 | ndmfv 6702 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅) | |
14 | 12, 13 | sylnbir 333 | . . . 4 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅) |
15 | 14 | fveq2d 6676 | . . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘∅)) |
16 | 9, 15, 14 | 3eqtr4a 2884 | . 2 ⊢ (¬ 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴)) |
17 | 8, 16 | pm2.61i 184 | 1 ⊢ (cf‘(cf‘𝐴)) = (cf‘𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1083 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 ⊆ wss 3938 ∅c0 4293 dom cdm 5557 Oncon0 6193 ⟶wf 6353 ‘cfv 6357 Smo wsmo 7984 cfccf 9368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-smo 7985 df-recs 8010 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-card 9370 df-cf 9372 df-acn 9373 |
This theorem is referenced by: (None) |
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