Theorem cfcof 9696

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:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → (cf‘𝐴) = (cf‘𝐵)))

Distinct variable groups:   𝑤,𝑓,𝑧,𝐴   𝐵,𝑓,𝑤,𝑧

Proof of Theorem cfcof

Dummy variables 𝑐 𝑔 ℎ 𝑘 𝑟 𝑠 𝑡 𝑥 𝑦 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cfcoflem 9694	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → (cf‘𝐴) ⊆ (cf‘𝐵)))
2	1	imp 409	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤))) → (cf‘𝐴) ⊆ (cf‘𝐵))
3		cff1 9680	. . . . . . 7 ⊢ (𝐴 ∈ On → ∃𝑔(𝑔:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑠 ∈ 𝐴 ∃𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔‘𝑡)))
4		f1f 6575	. . . . . . . . 9 ⊢ (𝑔:(cf‘𝐴)–1-1→𝐴 → 𝑔:(cf‘𝐴)⟶𝐴)
5	4	anim1i 616	. . . . . . . 8 ⊢ ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑠 ∈ 𝐴 ∃𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔‘𝑡)) → (𝑔:(cf‘𝐴)⟶𝐴 ∧ ∀𝑠 ∈ 𝐴 ∃𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔‘𝑡)))
6	5	eximi 1835	. . . . . . 7 ⊢ (∃𝑔(𝑔:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑠 ∈ 𝐴 ∃𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔‘𝑡)) → ∃𝑔(𝑔:(cf‘𝐴)⟶𝐴 ∧ ∀𝑠 ∈ 𝐴 ∃𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔‘𝑡)))
7	3, 6	syl 17	. . . . . 6 ⊢ (𝐴 ∈ On → ∃𝑔(𝑔:(cf‘𝐴)⟶𝐴 ∧ ∀𝑠 ∈ 𝐴 ∃𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔‘𝑡)))
8		eqid 2821	. . . . . . 7 ⊢ (𝑦 ∈ (cf‘𝐴) ↦ ∩ {𝑣 ∈ 𝐵 ∣ (𝑔‘𝑦) ⊆ (𝑓‘𝑣)}) = (𝑦 ∈ (cf‘𝐴) ↦ ∩ {𝑣 ∈ 𝐵 ∣ (𝑔‘𝑦) ⊆ (𝑓‘𝑣)})
9	8	coftr 9695	. . . . . 6 ⊢ (∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → (∃𝑔(𝑔:(cf‘𝐴)⟶𝐴 ∧ ∀𝑠 ∈ 𝐴 ∃𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔‘𝑡)) → ∃ℎ(ℎ:(cf‘𝐴)⟶𝐵 ∧ ∀𝑟 ∈ 𝐵 ∃𝑡 ∈ (cf‘𝐴)𝑟 ⊆ (ℎ‘𝑡))))
10	7, 9	syl5com 31	. . . . 5 ⊢ (𝐴 ∈ On → (∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → ∃ℎ(ℎ:(cf‘𝐴)⟶𝐵 ∧ ∀𝑟 ∈ 𝐵 ∃𝑡 ∈ (cf‘𝐴)𝑟 ⊆ (ℎ‘𝑡))))
11		eloni 6201	. . . . . . 7 ⊢ (𝐵 ∈ On → Ord 𝐵)
12		cfon 9677	. . . . . . 7 ⊢ (cf‘𝐴) ∈ On
13		eqid 2821	. . . . . . . 8 ⊢ {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡 ∈ 𝑥 (ℎ‘𝑡) ∈ (ℎ‘𝑥)} = {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡 ∈ 𝑥 (ℎ‘𝑡) ∈ (ℎ‘𝑥)}
14		eqid 2821	. . . . . . . 8 ⊢ ∩ {𝑐 ∈ (cf‘𝐴) ∣ 𝑟 ⊆ (ℎ‘𝑐)} = ∩ {𝑐 ∈ (cf‘𝐴) ∣ 𝑟 ⊆ (ℎ‘𝑐)}
15		eqid 2821	. . . . . . . 8 ⊢ OrdIso( E , {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡 ∈ 𝑥 (ℎ‘𝑡) ∈ (ℎ‘𝑥)}) = OrdIso( E , {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡 ∈ 𝑥 (ℎ‘𝑡) ∈ (ℎ‘𝑥)})
16	13, 14, 15	cofsmo 9691	. . . . . . 7 ⊢ ((Ord 𝐵 ∧ (cf‘𝐴) ∈ On) → (∃ℎ(ℎ:(cf‘𝐴)⟶𝐵 ∧ ∀𝑟 ∈ 𝐵 ∃𝑡 ∈ (cf‘𝐴)𝑟 ⊆ (ℎ‘𝑡)) → ∃𝑐 ∈ suc (cf‘𝐴)∃𝑘(𝑘:𝑐⟶𝐵 ∧ Smo 𝑘 ∧ ∀𝑟 ∈ 𝐵 ∃𝑠 ∈ 𝑐 𝑟 ⊆ (𝑘‘𝑠))))
17	11, 12, 16	sylancl 588	. . . . . 6 ⊢ (𝐵 ∈ On → (∃ℎ(ℎ:(cf‘𝐴)⟶𝐵 ∧ ∀𝑟 ∈ 𝐵 ∃𝑡 ∈ (cf‘𝐴)𝑟 ⊆ (ℎ‘𝑡)) → ∃𝑐 ∈ suc (cf‘𝐴)∃𝑘(𝑘:𝑐⟶𝐵 ∧ Smo 𝑘 ∧ ∀𝑟 ∈ 𝐵 ∃𝑠 ∈ 𝑐 𝑟 ⊆ (𝑘‘𝑠))))
18	12	onsuci 7553	. . . . . . . . . . 11 ⊢ suc (cf‘𝐴) ∈ On
19	18	oneli 6298	. . . . . . . . . 10 ⊢ (𝑐 ∈ suc (cf‘𝐴) → 𝑐 ∈ On)
20		cfflb 9681	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑐 ∈ On) → (∃𝑘(𝑘:𝑐⟶𝐵 ∧ ∀𝑟 ∈ 𝐵 ∃𝑠 ∈ 𝑐 𝑟 ⊆ (𝑘‘𝑠)) → (cf‘𝐵) ⊆ 𝑐))
21	19, 20	sylan2 594	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) → (∃𝑘(𝑘:𝑐⟶𝐵 ∧ ∀𝑟 ∈ 𝐵 ∃𝑠 ∈ 𝑐 𝑟 ⊆ (𝑘‘𝑠)) → (cf‘𝐵) ⊆ 𝑐))
22		3simpb 1145	. . . . . . . . . 10 ⊢ ((𝑘:𝑐⟶𝐵 ∧ Smo 𝑘 ∧ ∀𝑟 ∈ 𝐵 ∃𝑠 ∈ 𝑐 𝑟 ⊆ (𝑘‘𝑠)) → (𝑘:𝑐⟶𝐵 ∧ ∀𝑟 ∈ 𝐵 ∃𝑠 ∈ 𝑐 𝑟 ⊆ (𝑘‘𝑠)))
23	22	eximi 1835	. . . . . . . . 9 ⊢ (∃𝑘(𝑘:𝑐⟶𝐵 ∧ Smo 𝑘 ∧ ∀𝑟 ∈ 𝐵 ∃𝑠 ∈ 𝑐 𝑟 ⊆ (𝑘‘𝑠)) → ∃𝑘(𝑘:𝑐⟶𝐵 ∧ ∀𝑟 ∈ 𝐵 ∃𝑠 ∈ 𝑐 𝑟 ⊆ (𝑘‘𝑠)))
24	21, 23	impel 508	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) ∧ ∃𝑘(𝑘:𝑐⟶𝐵 ∧ Smo 𝑘 ∧ ∀𝑟 ∈ 𝐵 ∃𝑠 ∈ 𝑐 𝑟 ⊆ (𝑘‘𝑠))) → (cf‘𝐵) ⊆ 𝑐)
25		onsssuc 6278	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ On ∧ (cf‘𝐴) ∈ On) → (𝑐 ⊆ (cf‘𝐴) ↔ 𝑐 ∈ suc (cf‘𝐴)))
26	19, 12, 25	sylancl 588	. . . . . . . . . 10 ⊢ (𝑐 ∈ suc (cf‘𝐴) → (𝑐 ⊆ (cf‘𝐴) ↔ 𝑐 ∈ suc (cf‘𝐴)))
27	26	ibir 270	. . . . . . . . 9 ⊢ (𝑐 ∈ suc (cf‘𝐴) → 𝑐 ⊆ (cf‘𝐴))
28	27	ad2antlr 725	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) ∧ ∃𝑘(𝑘:𝑐⟶𝐵 ∧ Smo 𝑘 ∧ ∀𝑟 ∈ 𝐵 ∃𝑠 ∈ 𝑐 𝑟 ⊆ (𝑘‘𝑠))) → 𝑐 ⊆ (cf‘𝐴))
29	24, 28	sstrd 3977	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) ∧ ∃𝑘(𝑘:𝑐⟶𝐵 ∧ Smo 𝑘 ∧ ∀𝑟 ∈ 𝐵 ∃𝑠 ∈ 𝑐 𝑟 ⊆ (𝑘‘𝑠))) → (cf‘𝐵) ⊆ (cf‘𝐴))
30	29	rexlimdva2 3287	. . . . . 6 ⊢ (𝐵 ∈ On → (∃𝑐 ∈ suc (cf‘𝐴)∃𝑘(𝑘:𝑐⟶𝐵 ∧ Smo 𝑘 ∧ ∀𝑟 ∈ 𝐵 ∃𝑠 ∈ 𝑐 𝑟 ⊆ (𝑘‘𝑠)) → (cf‘𝐵) ⊆ (cf‘𝐴)))
31	17, 30	syld 47	. . . . 5 ⊢ (𝐵 ∈ On → (∃ℎ(ℎ:(cf‘𝐴)⟶𝐵 ∧ ∀𝑟 ∈ 𝐵 ∃𝑡 ∈ (cf‘𝐴)𝑟 ⊆ (ℎ‘𝑡)) → (cf‘𝐵) ⊆ (cf‘𝐴)))
32	10, 31	sylan9 510	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → (cf‘𝐵) ⊆ (cf‘𝐴)))
33	32	imp 409	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤))) → (cf‘𝐵) ⊆ (cf‘𝐴))
34	2, 33	eqssd 3984	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤))) → (cf‘𝐴) = (cf‘𝐵))
35	34	ex 415	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → (cf‘𝐴) = (cf‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139  {crab 3142   ⊆ wss 3936  ∩ cint 4876   ↦ cmpt 5146   E cep 5464  Ord word 6190  Oncon0 6191  suc csuc 6193  ⟶wf 6351  –1-1→wf1 6352  ‘cfv 6355  Smo wsmo 7982  OrdIsocoi 8973  cfccf 9366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-smo 7983  df-recs 8008  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-oi 8974  df-card 9368  df-cf 9370  df-acn 9371

This theorem is referenced by:  alephsing  9698