Theorem cfcof 10271

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 10269	. . . 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 10255	. . . . . . 7 ⊢ (𝐴 ∈ On → ∃𝑔(𝑔:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑠 ∈ 𝐴 ∃𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔‘𝑡)))
4		f1f 6787	. . . . . . . . 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 2732	. . . . . . 7 ⊢ (𝑦 ∈ (cf‘𝐴) ↦ ∩ {𝑣 ∈ 𝐵 ∣ (𝑔‘𝑦) ⊆ (𝑓‘𝑣)}) = (𝑦 ∈ (cf‘𝐴) ↦ ∩ {𝑣 ∈ 𝐵 ∣ (𝑔‘𝑦) ⊆ (𝑓‘𝑣)})
9	8	coftr 10270	. . . . . 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 6374	. . . . . . 7 ⊢ (𝐵 ∈ On → Ord 𝐵)
12		cfon 10252	. . . . . . 7 ⊢ (cf‘𝐴) ∈ On
13		eqid 2732	. . . . . . . 8 ⊢ {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡 ∈ 𝑥 (ℎ‘𝑡) ∈ (ℎ‘𝑥)} = {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡 ∈ 𝑥 (ℎ‘𝑡) ∈ (ℎ‘𝑥)}
14		eqid 2732	. . . . . . . 8 ⊢ ∩ {𝑐 ∈ (cf‘𝐴) ∣ 𝑟 ⊆ (ℎ‘𝑐)} = ∩ {𝑐 ∈ (cf‘𝐴) ∣ 𝑟 ⊆ (ℎ‘𝑐)}
15		eqid 2732	. . . . . . . 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 10266	. . . . . . 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 7829	. . . . . . . . . . 11 ⊢ suc (cf‘𝐴) ∈ On
19	18	oneli 6478	. . . . . . . . . 10 ⊢ (𝑐 ∈ suc (cf‘𝐴) → 𝑐 ∈ On)
20		cfflb 10256	. . . . . . . . . 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 1149	. . . . . . . . . 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 6454	. . . . . . . . . . 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 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 3992	. . . . . . 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 3157	. . . . . 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 3999	. 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 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  {crab 3432   ⊆ wss 3948  ∩ cint 4950   ↦ cmpt 5231   E cep 5579  Ord word 6363  Oncon0 6364  suc csuc 6366  ⟶wf 6539  –1-1→wf1 6540  ‘cfv 6543  Smo wsmo 8347  OrdIsocoi 9506  cfccf 9934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-smo 8348  df-recs 8373  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-oi 9507  df-card 9936  df-cf 9938  df-acn 9939

This theorem is referenced by:  alephsing  10273