Theorem cfcof 10168

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 10166	. . . 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 406	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤))) → (cf‘𝐴) ⊆ (cf‘𝐵))
3		cff1 10152	. . . . . . 7 ⊢ (𝐴 ∈ On → ∃𝑔(𝑔:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑠 ∈ 𝐴 ∃𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔‘𝑡)))
4		f1f 6720	. . . . . . . . 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 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 2729	. . . . . . 7 ⊢ (𝑦 ∈ (cf‘𝐴) ↦ ∩ {𝑣 ∈ 𝐵 ∣ (𝑔‘𝑦) ⊆ (𝑓‘𝑣)}) = (𝑦 ∈ (cf‘𝐴) ↦ ∩ {𝑣 ∈ 𝐵 ∣ (𝑔‘𝑦) ⊆ (𝑓‘𝑣)})
9	8	coftr 10167	. . . . . 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 6317	. . . . . . 7 ⊢ (𝐵 ∈ On → Ord 𝐵)
12		cfon 10149	. . . . . . 7 ⊢ (cf‘𝐴) ∈ On
13		eqid 2729	. . . . . . . 8 ⊢ {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡 ∈ 𝑥 (ℎ‘𝑡) ∈ (ℎ‘𝑥)} = {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡 ∈ 𝑥 (ℎ‘𝑡) ∈ (ℎ‘𝑥)}
14		eqid 2729	. . . . . . . 8 ⊢ ∩ {𝑐 ∈ (cf‘𝐴) ∣ 𝑟 ⊆ (ℎ‘𝑐)} = ∩ {𝑐 ∈ (cf‘𝐴) ∣ 𝑟 ⊆ (ℎ‘𝑐)}
15		eqid 2729	. . . . . . . 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 10163	. . . . . . 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 7772	. . . . . . . . . . 11 ⊢ suc (cf‘𝐴) ∈ On
19	18	oneli 6422	. . . . . . . . . 10 ⊢ (𝑐 ∈ suc (cf‘𝐴) → 𝑐 ∈ On)
20		cfflb 10153	. . . . . . . . . 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 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 505	. . . . . . . 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 6399	. . . . . . . . . . 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 268	. . . . . . . . 9 ⊢ (𝑐 ∈ suc (cf‘𝐴) → 𝑐 ⊆ (cf‘𝐴))
28	27	ad2antlr 727	. . . . . . . 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 3946	. . . . . . 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 3132	. . . . . 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 507	. . . 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 406	. . 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 3953	. 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 412	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3394   ⊆ wss 3903  ∩ cint 4896   ↦ cmpt 5173   E cep 5518  Ord word 6306  Oncon0 6307  suc csuc 6309  ⟶wf 6478  –1-1→wf1 6479  ‘cfv 6482  Smo wsmo 8268  OrdIsocoi 9401  cfccf 9833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-smo 8269  df-recs 8294  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-oi 9402  df-card 9835  df-cf 9837  df-acn 9838

This theorem is referenced by:  alephsing  10170