Theorem cfcof 10184

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 10182	. . . 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 10168	. . . . . . 7 ⊢ (𝐴 ∈ On → ∃𝑔(𝑔:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑠 ∈ 𝐴 ∃𝑡 ∈ (cf‘𝐴)𝑠 ⊆ (𝑔‘𝑡)))
4		f1f 6730	. . . . . . . . 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 1836	. . . . . . 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 2736	. . . . . . 7 ⊢ (𝑦 ∈ (cf‘𝐴) ↦ ∩ {𝑣 ∈ 𝐵 ∣ (𝑔‘𝑦) ⊆ (𝑓‘𝑣)}) = (𝑦 ∈ (cf‘𝐴) ↦ ∩ {𝑣 ∈ 𝐵 ∣ (𝑔‘𝑦) ⊆ (𝑓‘𝑣)})
9	8	coftr 10183	. . . . . 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 6327	. . . . . . 7 ⊢ (𝐵 ∈ On → Ord 𝐵)
12		cfon 10165	. . . . . . 7 ⊢ (cf‘𝐴) ∈ On
13		eqid 2736	. . . . . . . 8 ⊢ {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡 ∈ 𝑥 (ℎ‘𝑡) ∈ (ℎ‘𝑥)} = {𝑥 ∈ (cf‘𝐴) ∣ ∀𝑡 ∈ 𝑥 (ℎ‘𝑡) ∈ (ℎ‘𝑥)}
14		eqid 2736	. . . . . . . 8 ⊢ ∩ {𝑐 ∈ (cf‘𝐴) ∣ 𝑟 ⊆ (ℎ‘𝑐)} = ∩ {𝑐 ∈ (cf‘𝐴) ∣ 𝑟 ⊆ (ℎ‘𝑐)}
15		eqid 2736	. . . . . . . 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 10179	. . . . . . 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 7781	. . . . . . . . . . 11 ⊢ suc (cf‘𝐴) ∈ On
19	18	oneli 6432	. . . . . . . . . 10 ⊢ (𝑐 ∈ suc (cf‘𝐴) → 𝑐 ∈ On)
20		cfflb 10169	. . . . . . . . . 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 1836	. . . . . . . . 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 6409	. . . . . . . . . . 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 3944	. . . . . . 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 3139	. . . . . 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 3951	. 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 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  {crab 3399   ⊆ wss 3901  ∩ cint 4902   ↦ cmpt 5179   E cep 5523  Ord word 6316  Oncon0 6317  suc csuc 6319  ⟶wf 6488  –1-1→wf1 6489  ‘cfv 6492  Smo wsmo 8277  OrdIsocoi 9414  cfccf 9849

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-smo 8278  df-recs 8303  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-oi 9415  df-card 9851  df-cf 9853  df-acn 9854

This theorem is referenced by:  alephsing  10186