Theorem cff 10158

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 9853	. 2 ⊢ cf = (𝑥 ∈ On ↦ ∩ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))})
2		cardon 9856	. . . . . . 7 ⊢ (card‘𝑧) ∈ On
3		eleq1 2824	. . . . . . 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 4020	. . 3 ⊢ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))} ⊆ On
8		cflem 10155	. . . 4 ⊢ (𝑥 ∈ On → ∃𝑦∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣)))
9		abn0 4337	. . . 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 7740	. . 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 7057	1 ⊢ cf:On⟶On

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2714   ≠ wne 2932  ∀wral 3051  ∃wrex 3060   ⊆ wss 3901  ∅c0 4285  ∩ cint 4902  Oncon0 6317  ⟶wf 6488  ‘cfv 6492  cardccrd 9847  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-sep 5241  ax-nul 5251  ax-pr 5377

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-rab 3400  df-v 3442  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-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-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-ord 6320  df-on 6321  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-card 9851  df-cf 9853

This theorem is referenced by:  cfub  10159  cardcf  10162  cflecard  10163  cfle  10164  cflim2  10173  cfidm  10185