Theorem cfval 4906

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:   x,y,z,w,A

Proof of Theorem cfval

Step	Hyp	Ref	Expression
1		cflem 4905	. . 3 |- (A e. On -> E.xE.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)))
2		intexab 2731	. . 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 198	. 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 2083	. . . . . . . 8 |- (v = A -> (y (_ v <-> y (_ A))
5		raleq1 1786	. . . . . . . 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 628	. . . . . . 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 616	. . . . . 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	exbidv 1279	. . . . 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	abbidv 1577	. . . 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 2538	. . 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 4818	. . 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 3779	. 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 704	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 3   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  A.wral 1645  E.wrex 1646  Vcvv 1811   (_ wss 2047  |^|cint 2533  Oncon0 2948  ` cfv 3182  cardccrd 4813  cfccf 4815

This theorem is referenced by:  cfub 4908  cflim 4909  cardcf 4911  cflecard 4912  cfeq0 4914  cfsuc 4915  cfom 4916

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-int 2534  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-cf 4818

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