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Theorem cfcof 8979
 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 8977 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (cf‘𝐴) ⊆ (cf‘𝐵)))
21imp 444 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (cf‘𝐴) ⊆ (cf‘𝐵))
3 cff1 8963 . . . . . . 7 (𝐴 ∈ On → ∃𝑔(𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑠𝐴𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔𝑡)))
4 f1f 6014 . . . . . . . . 9 (𝑔:(cf‘𝐴)–1-1𝐴𝑔:(cf‘𝐴)⟶𝐴)
54anim1i 590 . . . . . . . 8 ((𝑔:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑠𝐴𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔𝑡)) → (𝑔:(cf‘𝐴)⟶𝐴 ∧ ∀𝑠𝐴𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔𝑡)))
65eximi 1752 . . . . . . 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 2610 . . . . . . 7 (𝑦 ∈ (cf‘𝐴) ↦ {𝑣𝐵 ∣ (𝑔𝑦) ⊆ (𝑓𝑣)}) = (𝑦 ∈ (cf‘𝐴) ↦ {𝑣𝐵 ∣ (𝑔𝑦) ⊆ (𝑓𝑣)})
98coftr 8978 . . . . . 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 5650 . . . . . . 7 (𝐵 ∈ On → Ord 𝐵)
12 cfon 8960 . . . . . . 7 (cf‘𝐴) ∈ On
13 eqid 2610 . . . . . . . 8 {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡𝑥 (𝑡) ∈ (𝑥)} = {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡𝑥 (𝑡) ∈ (𝑥)}
14 eqid 2610 . . . . . . . 8 {𝑐 ∈ (cf‘𝐴) ∣ 𝑟 ⊆ (𝑐)} = {𝑐 ∈ (cf‘𝐴) ∣ 𝑟 ⊆ (𝑐)}
15 eqid 2610 . . . . . . . 8 OrdIso( E , {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡𝑥 (𝑡) ∈ (𝑥)}) = OrdIso( E , {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡𝑥 (𝑡) ∈ (𝑥)})
1613, 14, 15cofsmo 8974 . . . . . . 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 693 . . . . . 6 (𝐵 ∈ On → (∃(:(cf‘𝐴)⟶𝐵 ∧ ∀𝑟𝐵𝑡 ∈ (cf‘𝐴)𝑟 ⊆ (𝑡)) → ∃𝑐 ∈ suc (cf‘𝐴)∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠))))
18 3simpb 1052 . . . . . . . . . . . 12 ((𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)) → (𝑘:𝑐𝐵 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)))
1918eximi 1752 . . . . . . . . . . 11 (∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)) → ∃𝑘(𝑘:𝑐𝐵 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)))
2012onsuci 6930 . . . . . . . . . . . . 13 suc (cf‘𝐴) ∈ On
2120oneli 5752 . . . . . . . . . . . 12 (𝑐 ∈ suc (cf‘𝐴) → 𝑐 ∈ On)
22 cfflb 8964 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝑐 ∈ On) → (∃𝑘(𝑘:𝑐𝐵 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)) → (cf‘𝐵) ⊆ 𝑐))
2321, 22sylan2 490 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) → (∃𝑘(𝑘:𝑐𝐵 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)) → (cf‘𝐵) ⊆ 𝑐))
2419, 23syl5 33 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) → (∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)) → (cf‘𝐵) ⊆ 𝑐))
2524imp 444 . . . . . . . . 9 (((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) ∧ ∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠))) → (cf‘𝐵) ⊆ 𝑐)
26 onsssuc 5730 . . . . . . . . . . . 12 ((𝑐 ∈ On ∧ (cf‘𝐴) ∈ On) → (𝑐 ⊆ (cf‘𝐴) ↔ 𝑐 ∈ suc (cf‘𝐴)))
2721, 12, 26sylancl 693 . . . . . . . . . . 11 (𝑐 ∈ suc (cf‘𝐴) → (𝑐 ⊆ (cf‘𝐴) ↔ 𝑐 ∈ suc (cf‘𝐴)))
2827ibir 256 . . . . . . . . . 10 (𝑐 ∈ suc (cf‘𝐴) → 𝑐 ⊆ (cf‘𝐴))
2928ad2antlr 759 . . . . . . . . 9 (((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) ∧ ∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠))) → 𝑐 ⊆ (cf‘𝐴))
3025, 29sstrd 3578 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) ∧ ∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠))) → (cf‘𝐵) ⊆ (cf‘𝐴))
3130exp31 628 . . . . . . 7 (𝐵 ∈ On → (𝑐 ∈ suc (cf‘𝐴) → (∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)) → (cf‘𝐵) ⊆ (cf‘𝐴))))
3231rexlimdv 3012 . . . . . 6 (𝐵 ∈ On → (∃𝑐 ∈ suc (cf‘𝐴)∃𝑘(𝑘:𝑐𝐵 ∧ Smo 𝑘 ∧ ∀𝑟𝐵𝑠𝑐 𝑟 ⊆ (𝑘𝑠)) → (cf‘𝐵) ⊆ (cf‘𝐴)))
3317, 32syld 46 . . . . 5 (𝐵 ∈ On → (∃(:(cf‘𝐴)⟶𝐵 ∧ ∀𝑟𝐵𝑡 ∈ (cf‘𝐴)𝑟 ⊆ (𝑡)) → (cf‘𝐵) ⊆ (cf‘𝐴)))
3410, 33sylan9 687 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (cf‘𝐵) ⊆ (cf‘𝐴)))
3534imp 444 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (cf‘𝐵) ⊆ (cf‘𝐴))
362, 35eqssd 3585 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (cf‘𝐴) = (cf‘𝐵))
3736ex 449 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900   ⊆ wss 3540  ∩ cint 4410   ↦ cmpt 4643   E cep 4947  Ord word 5639  Oncon0 5640  suc csuc 5642  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  Smo wsmo 7329  OrdIsocoi 8297  cfccf 8646 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-smo 7330  df-recs 7355  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-oi 8298  df-card 8648  df-cf 8650  df-acn 8651 This theorem is referenced by:  alephsing  8981
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