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 e. On -> (cf` A) = |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. 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 e. On -> E.xE.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)))
2		intexab 1987	. . 3 |- (E.xE.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) <-> |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} e. V)
3	1, 2	sylib 173	. 2 |- (A e. On -> |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} e. V)
4		sseq2 1522	. . . . . . . 8 |- (v = A -> (y (_ v <-> y (_ A))
5		raleq 1324	. . . . . . . 8 |- (v = A -> (A.z e. v E.w e. y z (_ w <-> A.z e. A E.w e. y z (_ w))
6	4, 5	anbi12d 476	. . . . . . 7 |- (v = A -> ((y (_ v /\ A.z e. v E.w e. y z (_ w) <-> (y (_ A /\ A.z e. A E.w e. y z (_ w)))
7	6	anbi2d 468	. . . . . 6 |- (v = A -> ((x = (card` y) /\ (y (_ v /\ A.z e. v E.w e. y z (_ w)) <-> (x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))))
8	7	biexdv 936	. . . . 5 |- (v = A -> (E.y(x = (card` y) /\ (y (_ v /\ A.z e. v E.w e. y z (_ w)) <-> E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))))
9	8	biabdv 1183	. . . 4 |- (v = A -> {x | E.y(x = (card` y) /\ (y (_ v /\ A.z e. v E.w e. y z (_ w))} = {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
10	9	inteqd 1970	. . 3 |- (v = A -> |^|{x | E.y(x = (card` y) /\ (y (_ v /\ A.z e. v E.w e. y z (_ w))} = |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
11		df-cf 3625	. . 3 |- cf = {<.v, u>. | (v e. On /\ u = |^|{x | E.y(x = (card` y) /\ (y (_ v /\ A.z e. v E.w e. y z (_ w))})}
12	10, 11	fvopab4g 2870	. 2 |- ((A e. On /\ |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} e. V) -> (cf` A) = |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
13	3, 12	mpdan 527	1 |- (A e. On -> (cf` A) = |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})

Syntax hints:   -> wi 2   /\ wa 196  E.wex 678  {cab 1090   = wceq 1091   e. wcel 1092  A.wral 1201  E.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

metamath.org