![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 10292 | . . . 4 ⊢ (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝐴 ∈ On → (cf‘(cf‘𝐴)) ⊆ (cf‘𝐴)) |
3 | cfsmo 10309 | . . . 4 ⊢ (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (cf‘𝐴)𝑥 ⊆ (𝑓‘𝑦))) | |
4 | cfon 10293 | . . . . 5 ⊢ (cf‘𝐴) ∈ On | |
5 | cfcoflem 10310 | . . . . 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 4013 | . 2 ⊢ (𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘𝐴)) |
9 | cf0 10289 | . . 3 ⊢ (cf‘∅) = ∅ | |
10 | cff 10286 | . . . . . . 7 ⊢ cf:On⟶On | |
11 | 10 | fdmi 6748 | . . . . . 6 ⊢ dom cf = On |
12 | 11 | eleq2i 2831 | . . . . 5 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On) |
13 | ndmfv 6942 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅) | |
14 | 12, 13 | sylnbir 331 | . . . 4 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅) |
15 | 14 | fveq2d 6911 | . . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘(cf‘𝐴)) = (cf‘∅)) |
16 | 9, 15, 14 | 3eqtr4a 2801 | . 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 1086 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 ∅c0 4339 dom cdm 5689 Oncon0 6386 ⟶wf 6559 ‘cfv 6563 Smo wsmo 8384 cfccf 9975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-smo 8385 df-recs 8410 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-card 9977 df-cf 9979 df-acn 9980 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |