Theorem cff 10317

Description: Cofinality is a function on the class of ordinal numbers to the class of cardinal numbers. (Contributed by Mario Carneiro, 15-Sep-2013.)

Assertion

Ref	Expression
cff	⊢ cf:On⟶On

Proof of Theorem cff

Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cf 10010	. 2 ⊢ cf = (𝑥 ∈ On ↦ ∩ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))})
2		cardon 10013	. . . . . . 7 ⊢ (card‘𝑧) ∈ On
3		eleq1 2832	. . . . . . 7 ⊢ (𝑦 = (card‘𝑧) → (𝑦 ∈ On ↔ (card‘𝑧) ∈ On))
4	2, 3	mpbiri 258	. . . . . 6 ⊢ (𝑦 = (card‘𝑧) → 𝑦 ∈ On)
5	4	adantr 480	. . . . 5 ⊢ ((𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣)) → 𝑦 ∈ On)
6	5	exlimiv 1929	. . . 4 ⊢ (∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣)) → 𝑦 ∈ On)
7	6	abssi 4093	. . 3 ⊢ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))} ⊆ On
8		cflem 10314	. . . 4 ⊢ (𝑥 ∈ On → ∃𝑦∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣)))
9		abn0 4408	. . . 4 ⊢ ({𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))} ≠ ∅ ↔ ∃𝑦∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣)))
10	8, 9	sylibr 234	. . 3 ⊢ (𝑥 ∈ On → {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))} ≠ ∅)
11		oninton 7831	. . 3 ⊢ (({𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))} ⊆ On ∧ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))} ≠ ∅) → ∩ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))} ∈ On)
12	7, 10, 11	sylancr 586	. 2 ⊢ (𝑥 ∈ On → ∩ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))} ∈ On)
13	1, 12	fmpti 7146	1 ⊢ cf:On⟶On

Syntax hints:   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717   ≠ wne 2946  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976  ∅c0 4352  ∩ cint 4970  Oncon0 6395  ⟶wf 6569  ‘cfv 6573  cardccrd 10004  cfccf 10006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-card 10008  df-cf 10010

This theorem is referenced by:  cfub  10318  cardcf  10321  cflecard  10322  cfle  10323  cflim2  10332  cfidm  10344