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Theorem cfcof 10314
Description: If there is a cofinal map from 𝐴 to 𝐵, then they have the same cofinality. This was used as Definition 11.1 of [TakeutiZaring] p. 100, who defines an equivalence relation cof (𝐴, 𝐵) and defines our cf(𝐵) as the minimum 𝐵 such that cof (𝐴, 𝐵). (Contributed by Mario Carneiro, 20-Mar-2013.)
Assertion
Ref Expression
cfcof ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (cf‘𝐴) = (cf‘𝐵)))
Distinct variable groups:   𝑤,𝑓,𝑧,𝐴   𝐵,𝑓,𝑤,𝑧

Proof of Theorem cfcof
Dummy variables 𝑐 𝑔 𝑘 𝑟 𝑠 𝑡 𝑥 𝑦 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cfcoflem 10312 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (cf‘𝐴) ⊆ (cf‘𝐵)))
21imp 406 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (cf‘𝐴) ⊆ (cf‘𝐵))
3 cff1 10298 . . . . . . 7 (𝐴 ∈ On → ∃𝑔(𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑠𝐴𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔𝑡)))
4 f1f 6804 . . . . . . . . 9 (𝑔:(cf‘𝐴)–1-1𝐴𝑔:(cf‘𝐴)⟶𝐴)
54anim1i 615 . . . . . . . 8 ((𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑠𝐴𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔𝑡)) → (𝑔:(cf‘𝐴)⟶𝐴 ∧ ∀𝑠𝐴𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔𝑡)))
65eximi 1835 . . . . . . 7 (∃𝑔(𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑠𝐴𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔𝑡)) → ∃𝑔(𝑔:(cf‘𝐴)⟶𝐴 ∧ ∀𝑠𝐴𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔𝑡)))
73, 6syl 17 . . . . . 6 (𝐴 ∈ On → ∃𝑔(𝑔:(cf‘𝐴)⟶𝐴 ∧ ∀𝑠𝐴𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔𝑡)))
8 eqid 2737 . . . . . . 7 (𝑦 ∈ (cf‘𝐴) ↦ {𝑣𝐵 ∣ (𝑔𝑦) ⊆ (𝑓𝑣)}) = (𝑦 ∈ (cf‘𝐴) ↦ {𝑣𝐵 ∣ (𝑔𝑦) ⊆ (𝑓𝑣)})
98coftr 10313 . . . . . 6 (∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (∃𝑔(𝑔:(cf‘𝐴)⟶𝐴 ∧ ∀𝑠𝐴𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔𝑡)) → ∃(:(cf‘𝐴)⟶𝐵 ∧ ∀𝑟𝐵𝑡 ∈ (cf‘𝐴)𝑟 ⊆ (𝑡))))
107, 9syl5com 31 . . . . 5 (𝐴 ∈ On → (∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ∃(:(cf‘𝐴)⟶𝐵 ∧ ∀𝑟𝐵𝑡 ∈ (cf‘𝐴)𝑟 ⊆ (𝑡))))
11 eloni 6394 . . . . . . 7 (𝐵 ∈ On → Ord 𝐵)
12 cfon 10295 . . . . . . 7 (cf‘𝐴) ∈ On
13 eqid 2737 . . . . . . . 8 {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡𝑥 (𝑡) ∈ (𝑥)} = {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡𝑥 (𝑡) ∈ (𝑥)}
14 eqid 2737 . . . . . . . 8 {𝑐 ∈ (cf‘𝐴) ∣ 𝑟 ⊆ (𝑐)} = {𝑐 ∈ (cf‘𝐴) ∣ 𝑟 ⊆ (𝑐)}
15 eqid 2737 . . . . . . . 8 OrdIso( E , {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡𝑥 (𝑡) ∈ (𝑥)}) = OrdIso( E , {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡𝑥 (𝑡) ∈ (𝑥)})
1613, 14, 15cofsmo 10309 . . . . . . 7 ((Ord 𝐵 ∧ (cf‘𝐴) ∈ On) → (∃(:(cf‘𝐴)⟶𝐵 ∧ ∀𝑟𝐵𝑡 ∈ (cf‘𝐴)𝑟 ⊆ (𝑡)) → ∃𝑐 ∈ suc (cf‘𝐴)∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠))))
1711, 12, 16sylancl 586 . . . . . 6 (𝐵 ∈ On → (∃(:(cf‘𝐴)⟶𝐵 ∧ ∀𝑟𝐵𝑡 ∈ (cf‘𝐴)𝑟 ⊆ (𝑡)) → ∃𝑐 ∈ suc (cf‘𝐴)∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠))))
1812onsuci 7859 . . . . . . . . . . 11 suc (cf‘𝐴) ∈ On
1918oneli 6498 . . . . . . . . . 10 (𝑐 ∈ suc (cf‘𝐴) → 𝑐 ∈ On)
20 cfflb 10299 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑐 ∈ On) → (∃𝑘(𝑘:𝑐𝐵 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)) → (cf‘𝐵) ⊆ 𝑐))
2119, 20sylan2 593 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) → (∃𝑘(𝑘:𝑐𝐵 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)) → (cf‘𝐵) ⊆ 𝑐))
22 3simpb 1150 . . . . . . . . . 10 ((𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)) → (𝑘:𝑐𝐵 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)))
2322eximi 1835 . . . . . . . . 9 (∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)) → ∃𝑘(𝑘:𝑐𝐵 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)))
2421, 23impel 505 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) ∧ ∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠))) → (cf‘𝐵) ⊆ 𝑐)
25 onsssuc 6474 . . . . . . . . . . 11 ((𝑐 ∈ On ∧ (cf‘𝐴) ∈ On) → (𝑐 ⊆ (cf‘𝐴) ↔ 𝑐 ∈ suc (cf‘𝐴)))
2619, 12, 25sylancl 586 . . . . . . . . . 10 (𝑐 ∈ suc (cf‘𝐴) → (𝑐 ⊆ (cf‘𝐴) ↔ 𝑐 ∈ suc (cf‘𝐴)))
2726ibir 268 . . . . . . . . 9 (𝑐 ∈ suc (cf‘𝐴) → 𝑐 ⊆ (cf‘𝐴))
2827ad2antlr 727 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) ∧ ∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠))) → 𝑐 ⊆ (cf‘𝐴))
2924, 28sstrd 3994 . . . . . . 7 (((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) ∧ ∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠))) → (cf‘𝐵) ⊆ (cf‘𝐴))
3029rexlimdva2 3157 . . . . . 6 (𝐵 ∈ On → (∃𝑐 ∈ suc (cf‘𝐴)∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)) → (cf‘𝐵) ⊆ (cf‘𝐴)))
3117, 30syld 47 . . . . 5 (𝐵 ∈ On → (∃(:(cf‘𝐴)⟶𝐵 ∧ ∀𝑟𝐵𝑡 ∈ (cf‘𝐴)𝑟 ⊆ (𝑡)) → (cf‘𝐵) ⊆ (cf‘𝐴)))
3210, 31sylan9 507 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (cf‘𝐵) ⊆ (cf‘𝐴)))
3332imp 406 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (cf‘𝐵) ⊆ (cf‘𝐴))
342, 33eqssd 4001 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (cf‘𝐴) = (cf‘𝐵))
3534ex 412 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (cf‘𝐴) = (cf‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wral 3061  wrex 3070  {crab 3436  wss 3951   cint 4946  cmpt 5225   E cep 5583  Ord word 6383  Oncon0 6384  suc csuc 6386  wf 6557  1-1wf1 6558  cfv 6561  Smo wsmo 8385  OrdIsocoi 9549  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-lim 6389  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-oi 9550  df-card 9979  df-cf 9981  df-acn 9982
This theorem is referenced by:  alephsing  10316
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