Theorem cfcof 10030

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 10028	. . . 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 407	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤))) → (cf‘𝐴) ⊆ (cf‘𝐵))
3		cff1 10014	. . . . . . 7 ⊢ (𝐴 ∈ On → ∃𝑔(𝑔:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑠 ∈ 𝐴 ∃𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔‘𝑡)))
4		f1f 6670	. . . . . . . . 9 ⊢ (𝑔:(cf‘𝐴)–1-1→𝐴 → 𝑔:(cf‘𝐴)⟶𝐴)
5	4	anim1i 615	. . . . . . . 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 1837	. . . . . . 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 2738	. . . . . . 7 ⊢ (𝑦 ∈ (cf‘𝐴) ↦ ∩ {𝑣 ∈ 𝐵 ∣ (𝑔‘𝑦) ⊆ (𝑓‘𝑣)}) = (𝑦 ∈ (cf‘𝐴) ↦ ∩ {𝑣 ∈ 𝐵 ∣ (𝑔‘𝑦) ⊆ (𝑓‘𝑣)})
9	8	coftr 10029	. . . . . 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 6276	. . . . . . 7 ⊢ (𝐵 ∈ On → Ord 𝐵)
12		cfon 10011	. . . . . . 7 ⊢ (cf‘𝐴) ∈ On
13		eqid 2738	. . . . . . . 8 ⊢ {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡 ∈ 𝑥 (ℎ‘𝑡) ∈ (ℎ‘𝑥)} = {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡 ∈ 𝑥 (ℎ‘𝑡) ∈ (ℎ‘𝑥)}
14		eqid 2738	. . . . . . . 8 ⊢ ∩ {𝑐 ∈ (cf‘𝐴) ∣ 𝑟 ⊆ (ℎ‘𝑐)} = ∩ {𝑐 ∈ (cf‘𝐴) ∣ 𝑟 ⊆ (ℎ‘𝑐)}
15		eqid 2738	. . . . . . . 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 10025	. . . . . . 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 586	. . . . . 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 7685	. . . . . . . . . . 11 ⊢ suc (cf‘𝐴) ∈ On
19	18	oneli 6374	. . . . . . . . . 10 ⊢ (𝑐 ∈ suc (cf‘𝐴) → 𝑐 ∈ On)
20		cfflb 10015	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑐 ∈ On) → (∃𝑘(𝑘:𝑐⟶𝐵 ∧ ∀𝑟 ∈ 𝐵 ∃𝑠 ∈ 𝑐 𝑟 ⊆ (𝑘‘𝑠)) → (cf‘𝐵) ⊆ 𝑐))
21	19, 20	sylan2 593	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝑐 ∈ suc (cf‘𝐴)) → (∃𝑘(𝑘:𝑐⟶𝐵 ∧ ∀𝑟 ∈ 𝐵 ∃𝑠 ∈ 𝑐 𝑟 ⊆ (𝑘‘𝑠)) → (cf‘𝐵) ⊆ 𝑐))
22		3simpb 1148	. . . . . . . . . 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 1837	. . . . . . . . 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 506	. . . . . . . 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 6353	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ On ∧ (cf‘𝐴) ∈ On) → (𝑐 ⊆ (cf‘𝐴) ↔ 𝑐 ∈ suc (cf‘𝐴)))
26	19, 12, 25	sylancl 586	. . . . . . . . . 10 ⊢ (𝑐 ∈ suc (cf‘𝐴) → (𝑐 ⊆ (cf‘𝐴) ↔ 𝑐 ∈ suc (cf‘𝐴)))
27	26	ibir 267	. . . . . . . . 9 ⊢ (𝑐 ∈ suc (cf‘𝐴) → 𝑐 ⊆ (cf‘𝐴))
28	27	ad2antlr 724	. . . . . . . 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 3931	. . . . . . 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 3216	. . . . . 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 508	. . . 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 407	. . 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 3938	. 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 413	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 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  {crab 3068   ⊆ wss 3887  ∩ cint 4879   ↦ cmpt 5157   E cep 5494  Ord word 6265  Oncon0 6266  suc csuc 6268  ⟶wf 6429  –1-1→wf1 6430  ‘cfv 6433  Smo wsmo 8176  OrdIsocoi 9268  cfccf 9695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-smo 8177  df-recs 8202  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-oi 9269  df-card 9697  df-cf 9699  df-acn 9700

This theorem is referenced by:  alephsing  10032