Theorem cff 10150

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 9845	. 2 ⊢ cf = (𝑥 ∈ On ↦ ∩ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))})
2		cardon 9848	. . . . . . 7 ⊢ (card‘𝑧) ∈ On
3		eleq1 2821	. . . . . . 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 1931	. . . 4 ⊢ (∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣)) → 𝑦 ∈ On)
7	6	abssi 4017	. . 3 ⊢ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))} ⊆ On
8		cflem 10147	. . . 4 ⊢ (𝑥 ∈ On → ∃𝑦∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣)))
9		abn0 4334	. . . 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 7737	. . 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 587	. 2 ⊢ (𝑥 ∈ On → ∩ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))} ∈ On)
13	1, 12	fmpti 7054	1 ⊢ cf:On⟶On

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2711   ≠ wne 2929  ∀wral 3048  ∃wrex 3057   ⊆ wss 3898  ∅c0 4282  ∩ cint 4899  Oncon0 6314  ⟶wf 6485  ‘cfv 6489  cardccrd 9839  cfccf 9841

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 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6317  df-on 6318  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497  df-card 9843  df-cf 9845

This theorem is referenced by:  cfub  10151  cardcf  10154  cflecard  10155  cfle  10156  cflim2  10165  cfidm  10177