Theorem cff 10227

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 9923	. 2 ⊢ cf = (𝑥 ∈ On ↦ ∩ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))})
2		cardon 9926	. . . . . . 7 ⊢ (card‘𝑧) ∈ On
3		eleq1 2857	. . . . . . 7 ⊢ (𝑦 = (card‘𝑧) → (𝑦 ∈ On ↔ (card‘𝑧) ∈ On))
4	2, 3	mpbiri 261	. . . . . 6 ⊢ (𝑦 = (card‘𝑧) → 𝑦 ∈ On)
5	4	adantr 485	. . . . 5 ⊢ ((𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣)) → 𝑦 ∈ On)
6	5	exlimiv 1957	. . . 4 ⊢ (∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣)) → 𝑦 ∈ On)
7	6	abssi 4030	. . 3 ⊢ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))} ⊆ On
8		cflem 10224	. . . 4 ⊢ (𝑥 ∈ On → ∃𝑦∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣)))
9		abn0 4347	. . . 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 237	. . 3 ⊢ (𝑥 ∈ On → {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))} ≠ ∅)
11		oninton 7790	. . 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 598	. 2 ⊢ (𝑥 ∈ On → ∩ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))} ∈ On)
13	1, 12	fmpti 7105	1 ⊢ cf:On⟶On

Syntax hints:   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   ⊆ wss 3913  ∅c0 4294  ∩ cint 4913  Oncon0 6358  ⟶wf 6530  ‘cfv 6534  cardccrd 9917  cfccf 9919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6361  df-on 6362  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-card 9921  df-cf 9923

This theorem is referenced by:  cfub  10228  cardcf  10231  cflecard  10232  cfle  10233  cfon  10234  cflim2  10243  cfidm  10255