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Theorem cfcof 10312
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 10310 . . . 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 10296 . . . . . . 7 (𝐴 ∈ On → ∃𝑔(𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑠𝐴𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔𝑡)))
4 f1f 6805 . . . . . . . . 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 1832 . . . . . . 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 2735 . . . . . . 7 (𝑦 ∈ (cf‘𝐴) ↦ {𝑣𝐵 ∣ (𝑔𝑦) ⊆ (𝑓𝑣)}) = (𝑦 ∈ (cf‘𝐴) ↦ {𝑣𝐵 ∣ (𝑔𝑦) ⊆ (𝑓𝑣)})
98coftr 10311 . . . . . 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 6396 . . . . . . 7 (𝐵 ∈ On → Ord 𝐵)
12 cfon 10293 . . . . . . 7 (cf‘𝐴) ∈ On
13 eqid 2735 . . . . . . . 8 {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡𝑥 (𝑡) ∈ (𝑥)} = {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡𝑥 (𝑡) ∈ (𝑥)}
14 eqid 2735 . . . . . . . 8 {𝑐 ∈ (cf‘𝐴) ∣ 𝑟 ⊆ (𝑐)} = {𝑐 ∈ (cf‘𝐴) ∣ 𝑟 ⊆ (𝑐)}
15 eqid 2735 . . . . . . . 8 OrdIso( E , {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡𝑥 (𝑡) ∈ (𝑥)}) = OrdIso( E , {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡𝑥 (𝑡) ∈ (𝑥)})
1613, 14, 15cofsmo 10307 . . . . . . 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 6500 . . . . . . . . . 10 (𝑐 ∈ suc (cf‘𝐴) → 𝑐 ∈ On)
20 cfflb 10297 . . . . . . . . . 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 1148 . . . . . . . . . 10 ((𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)) → (𝑘:𝑐𝐵 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)))
2322eximi 1832 . . . . . . . . 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 6476 . . . . . . . . . . 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 4006 . . . . . . 7 (((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) ∧ ∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠))) → (cf‘𝐵) ⊆ (cf‘𝐴))
3029rexlimdva2 3155 . . . . . 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 4013 . 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 1086   = wceq 1537  wex 1776  wcel 2106  wral 3059  wrex 3068  {crab 3433  wss 3963   cint 4951  cmpt 5231   E cep 5588  Ord word 6385  Oncon0 6386  suc csuc 6388  wf 6559  1-1wf1 6560  cfv 6563  Smo wsmo 8384  OrdIsocoi 9547  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-lim 6391  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-oi 9548  df-card 9977  df-cf 9979  df-acn 9980
This theorem is referenced by:  alephsing  10314
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