Theorem cfval 3701

Description: Value of the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). The cofinality of an ordinal number A is the cardinality (size) of the smallest unbounded subset y of the ordinal number. Unbounded means that for every member of A, there is a member of y that is at least as large. Cofinality is a measure of how "reachable from below" an ordinal is.

Assertion

Ref	Expression
cfval	⊢ (A ∈ On → (cf ‘A) = ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))})

Distinct variable group(s):   x,y,z,w,A

Proof of Theorem cfval

Step	Hyp	Ref	Expression
1		cflem 3700	. . 3 ⊢ (A ∈ On → ∃x∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w)))
2		intexab 1987	. . 3 ⊢ (∃x∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w)) ↔ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))} ∈ V)
3	1, 2	sylib 173	. 2 ⊢ (A ∈ On → ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))} ∈ V)
4		sseq2 1522	. . . . . . . 8 ⊢ (v = A → (y ⊆ v ↔ y ⊆ A))
5		raleq 1324	. . . . . . . 8 ⊢ (v = A → (∀z ∈ v ∃w ∈ y z ⊆ w ↔ ∀z ∈ A ∃w ∈ y z ⊆ w))
6	4, 5	anbi12d 476	. . . . . . 7 ⊢ (v = A → ((y ⊆ v ∧ ∀z ∈ v ∃w ∈ y z ⊆ w) ↔ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w)))
7	6	anbi2d 468	. . . . . 6 ⊢ (v = A → ((x = (card ‘y) ∧ (y ⊆ v ∧ ∀z ∈ v ∃w ∈ y z ⊆ w)) ↔ (x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))))
8	7	biexdv 936	. . . . 5 ⊢ (v = A → (∃y(x = (card ‘y) ∧ (y ⊆ v ∧ ∀z ∈ v ∃w ∈ y z ⊆ w)) ↔ ∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))))
9	8	biabdv 1183	. . . 4 ⊢ (v = A → {x∣∃y(x = (card ‘y) ∧ (y ⊆ v ∧ ∀z ∈ v ∃w ∈ y z ⊆ w))} = {x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))})
10	9	inteqd 1970	. . 3 ⊢ (v = A → ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ v ∧ ∀z ∈ v ∃w ∈ y z ⊆ w))} = ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))})
11		df-cf 3625	. . 3 ⊢ cf = {⟨v, u⟩∣(v ∈ On ∧ u = ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ v ∧ ∀z ∈ v ∃w ∈ y z ⊆ w))})}
12	10, 11	fvopab4g 2870	. 2 ⊢ ((A ∈ On ∧ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))} ∈ V) → (cf ‘A) = ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))})
13	3, 12	mpdan 527	1 ⊢ (A ∈ On → (cf ‘A) = ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))})

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  Vcvv 1348   ⊆ wss 1487  ∩cint 1965  Oncon0 2199   ‘cfv 2422  cardccrd 3620  cfccf 3622

This theorem is referenced by:  cfub 3703  cflim 3704  cardcf 3706  cflecard 3707  cfsuc 3709  cfom 3710

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-int 1966  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-cf 3625

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