Theorem cfval 9069

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. (Contributed by NM, 1-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)

Assertion

Ref	Expression
cfval	|- ( A e. On -> ( cf ` A ) = |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } )

Distinct variable group:   x, y, z, w, A

Proof of Theorem cfval

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cflem 9068	. . 3 |- ( A e. On -> E. x E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) )
2		intexab 4822	. . 3 |- ( E. x E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) <-> |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } e. _V )
3	1, 2	sylib 208	. 2 |- ( A e. On -> |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } e. _V )
4		sseq2 3627	. . . . . . . 8 |- ( v = A -> ( y C_ v <-> y C_ A ) )
5		raleq 3138	. . . . . . . 8 |- ( v = A -> ( A. z e. v E. w e. y z C_ w <-> A. z e. A E. w e. y z C_ w ) )
6	4, 5	anbi12d 747	. . . . . . 7 |- ( v = A -> ( ( y C_ v /\ A. z e. v E. w e. y z C_ w ) <-> ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) )
7	6	anbi2d 740	. . . . . 6 |- ( v = A -> ( ( x = ( card ` y ) /\ ( y C_ v /\ A. z e. v E. w e. y z C_ w ) ) <-> ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) ) )
8	7	exbidv 1850	. . . . 5 |- ( v = A -> ( E. y ( x = ( card ` y ) /\ ( y C_ v /\ A. z e. v E. w e. y z C_ w ) ) <-> E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) ) )
9	8	abbidv 2741	. . . 4 |- ( v = A -> { x | E. y ( x = ( card ` y ) /\ ( y C_ v /\ A. z e. v E. w e. y z C_ w ) ) } = { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } )
10	9	inteqd 4480	. . 3 |- ( v = A -> |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ v /\ A. z e. v E. w e. y z C_ w ) ) } = |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } )
11		df-cf 8767	. . 3 |- cf = ( v e. On |-> |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ v /\ A. z e. v E. w e. y z C_ w ) ) } )
12	10, 11	fvmptg 6280	. 2 |- ( ( A e. On /\ |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } e. _V ) -> ( cf ` A ) = |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } )
13	3, 12	mpdan 702	1 |- ( A e. On -> ( cf ` A ) = |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   |^|cint 4475   Oncon0 5723   ` cfv 5888   cardccrd 8761   cfccf 8763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-cf 8767

This theorem is referenced by:  cfub  9071  cflm  9072  cardcf  9074  cflecard  9075  cfeq0  9078  cfsuc  9079  cff1  9080  cfflb  9081  cfval2  9082  cflim3  9084