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Theorem cfcof 10317
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 10315 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (cf‘𝐴) ⊆ (cf‘𝐵)))
21imp 405 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (cf‘𝐴) ⊆ (cf‘𝐵))
3 cff1 10301 . . . . . . 7 (𝐴 ∈ On → ∃𝑔(𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑠𝐴𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔𝑡)))
4 f1f 6798 . . . . . . . . 9 (𝑔:(cf‘𝐴)–1-1𝐴𝑔:(cf‘𝐴)⟶𝐴)
54anim1i 613 . . . . . . . 8 ((𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑠𝐴𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔𝑡)) → (𝑔:(cf‘𝐴)⟶𝐴 ∧ ∀𝑠𝐴𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔𝑡)))
65eximi 1830 . . . . . . 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 2726 . . . . . . 7 (𝑦 ∈ (cf‘𝐴) ↦ {𝑣𝐵 ∣ (𝑔𝑦) ⊆ (𝑓𝑣)}) = (𝑦 ∈ (cf‘𝐴) ↦ {𝑣𝐵 ∣ (𝑔𝑦) ⊆ (𝑓𝑣)})
98coftr 10316 . . . . . 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 6386 . . . . . . 7 (𝐵 ∈ On → Ord 𝐵)
12 cfon 10298 . . . . . . 7 (cf‘𝐴) ∈ On
13 eqid 2726 . . . . . . . 8 {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡𝑥 (𝑡) ∈ (𝑥)} = {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡𝑥 (𝑡) ∈ (𝑥)}
14 eqid 2726 . . . . . . . 8 {𝑐 ∈ (cf‘𝐴) ∣ 𝑟 ⊆ (𝑐)} = {𝑐 ∈ (cf‘𝐴) ∣ 𝑟 ⊆ (𝑐)}
15 eqid 2726 . . . . . . . 8 OrdIso( E , {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡𝑥 (𝑡) ∈ (𝑥)}) = OrdIso( E , {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡𝑥 (𝑡) ∈ (𝑥)})
1613, 14, 15cofsmo 10312 . . . . . . 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 584 . . . . . 6 (𝐵 ∈ On → (∃(:(cf‘𝐴)⟶𝐵 ∧ ∀𝑟𝐵𝑡 ∈ (cf‘𝐴)𝑟 ⊆ (𝑡)) → ∃𝑐 ∈ suc (cf‘𝐴)∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠))))
1812onsuci 7848 . . . . . . . . . . 11 suc (cf‘𝐴) ∈ On
1918oneli 6490 . . . . . . . . . 10 (𝑐 ∈ suc (cf‘𝐴) → 𝑐 ∈ On)
20 cfflb 10302 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑐 ∈ On) → (∃𝑘(𝑘:𝑐𝐵 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)) → (cf‘𝐵) ⊆ 𝑐))
2119, 20sylan2 591 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) → (∃𝑘(𝑘:𝑐𝐵 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)) → (cf‘𝐵) ⊆ 𝑐))
22 3simpb 1146 . . . . . . . . . 10 ((𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)) → (𝑘:𝑐𝐵 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)))
2322eximi 1830 . . . . . . . . 9 (∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)) → ∃𝑘(𝑘:𝑐𝐵 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)))
2421, 23impel 504 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) ∧ ∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠))) → (cf‘𝐵) ⊆ 𝑐)
25 onsssuc 6466 . . . . . . . . . . 11 ((𝑐 ∈ On ∧ (cf‘𝐴) ∈ On) → (𝑐 ⊆ (cf‘𝐴) ↔ 𝑐 ∈ suc (cf‘𝐴)))
2619, 12, 25sylancl 584 . . . . . . . . . 10 (𝑐 ∈ suc (cf‘𝐴) → (𝑐 ⊆ (cf‘𝐴) ↔ 𝑐 ∈ suc (cf‘𝐴)))
2726ibir 267 . . . . . . . . 9 (𝑐 ∈ suc (cf‘𝐴) → 𝑐 ⊆ (cf‘𝐴))
2827ad2antlr 725 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) ∧ ∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠))) → 𝑐 ⊆ (cf‘𝐴))
2924, 28sstrd 3990 . . . . . . 7 (((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) ∧ ∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠))) → (cf‘𝐵) ⊆ (cf‘𝐴))
3029rexlimdva2 3147 . . . . . 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 506 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (cf‘𝐵) ⊆ (cf‘𝐴)))
3332imp 405 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (cf‘𝐵) ⊆ (cf‘𝐴))
342, 33eqssd 3997 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (cf‘𝐴) = (cf‘𝐵))
3534ex 411 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 205  wa 394  w3a 1084   = wceq 1534  wex 1774  wcel 2099  wral 3051  wrex 3060  {crab 3419  wss 3947   cint 4954  cmpt 5236   E cep 5585  Ord word 6375  Oncon0 6376  suc csuc 6378  wf 6550  1-1wf1 6551  cfv 6554  Smo wsmo 8375  OrdIsocoi 9552  cfccf 9980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-smo 8376  df-recs 8401  df-er 8734  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-oi 9553  df-card 9982  df-cf 9984  df-acn 9985
This theorem is referenced by:  alephsing  10319
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