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Theorem cfcof 9549
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 9547 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (cf‘𝐴) ⊆ (cf‘𝐵)))
21imp 407 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (cf‘𝐴) ⊆ (cf‘𝐵))
3 cff1 9533 . . . . . . 7 (𝐴 ∈ On → ∃𝑔(𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑠𝐴𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔𝑡)))
4 f1f 6450 . . . . . . . . 9 (𝑔:(cf‘𝐴)–1-1𝐴𝑔:(cf‘𝐴)⟶𝐴)
54anim1i 614 . . . . . . . 8 ((𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑠𝐴𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔𝑡)) → (𝑔:(cf‘𝐴)⟶𝐴 ∧ ∀𝑠𝐴𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔𝑡)))
65eximi 1820 . . . . . . 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 2797 . . . . . . 7 (𝑦 ∈ (cf‘𝐴) ↦ {𝑣𝐵 ∣ (𝑔𝑦) ⊆ (𝑓𝑣)}) = (𝑦 ∈ (cf‘𝐴) ↦ {𝑣𝐵 ∣ (𝑔𝑦) ⊆ (𝑓𝑣)})
98coftr 9548 . . . . . 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 6083 . . . . . . 7 (𝐵 ∈ On → Ord 𝐵)
12 cfon 9530 . . . . . . 7 (cf‘𝐴) ∈ On
13 eqid 2797 . . . . . . . 8 {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡𝑥 (𝑡) ∈ (𝑥)} = {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡𝑥 (𝑡) ∈ (𝑥)}
14 eqid 2797 . . . . . . . 8 {𝑐 ∈ (cf‘𝐴) ∣ 𝑟 ⊆ (𝑐)} = {𝑐 ∈ (cf‘𝐴) ∣ 𝑟 ⊆ (𝑐)}
15 eqid 2797 . . . . . . . 8 OrdIso( E , {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡𝑥 (𝑡) ∈ (𝑥)}) = OrdIso( E , {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡𝑥 (𝑡) ∈ (𝑥)})
1613, 14, 15cofsmo 9544 . . . . . . 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 7416 . . . . . . . . . . 11 suc (cf‘𝐴) ∈ On
1918oneli 6180 . . . . . . . . . 10 (𝑐 ∈ suc (cf‘𝐴) → 𝑐 ∈ On)
20 cfflb 9534 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑐 ∈ On) → (∃𝑘(𝑘:𝑐𝐵 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)) → (cf‘𝐵) ⊆ 𝑐))
2119, 20sylan2 592 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) → (∃𝑘(𝑘:𝑐𝐵 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)) → (cf‘𝐵) ⊆ 𝑐))
22 3simpb 1142 . . . . . . . . . 10 ((𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)) → (𝑘:𝑐𝐵 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)))
2322eximi 1820 . . . . . . . . 9 (∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)) → ∃𝑘(𝑘:𝑐𝐵 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)))
2421, 23impel 506 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) ∧ ∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠))) → (cf‘𝐵) ⊆ 𝑐)
25 onsssuc 6160 . . . . . . . . . . 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 269 . . . . . . . . 9 (𝑐 ∈ suc (cf‘𝐴) → 𝑐 ⊆ (cf‘𝐴))
2827ad2antlr 723 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) ∧ ∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠))) → 𝑐 ⊆ (cf‘𝐴))
2924, 28sstrd 3905 . . . . . . 7 (((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) ∧ ∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠))) → (cf‘𝐵) ⊆ (cf‘𝐴))
3029rexlimdva2 3252 . . . . . 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 508 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (cf‘𝐵) ⊆ (cf‘𝐴)))
3332imp 407 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (cf‘𝐵) ⊆ (cf‘𝐴))
342, 33eqssd 3912 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (cf‘𝐴) = (cf‘𝐵))
3534ex 413 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 207  wa 396  w3a 1080   = wceq 1525  wex 1765  wcel 2083  wral 3107  wrex 3108  {crab 3111  wss 3865   cint 4788  cmpt 5047   E cep 5359  Ord word 6072  Oncon0 6073  suc csuc 6075  wf 6228  1-1wf1 6229  cfv 6232  Smo wsmo 7841  OrdIsocoi 8826  cfccf 9219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-se 5410  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-smo 7842  df-recs 7867  df-er 8146  df-map 8265  df-en 8365  df-dom 8366  df-sdom 8367  df-oi 8827  df-card 9221  df-cf 9223  df-acn 9224
This theorem is referenced by:  alephsing  9551
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