Theorem cff 9070

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 8767	. 2 |- cf = ( x e. On |-> |^| { y | E. z ( y = ( card ` z ) /\ ( z C_ x /\ A. w e. x E. v e. z w C_ v ) ) } )
2		cardon 8770	. . . . . . 7 |- ( card ` z ) e. On
3		eleq1 2689	. . . . . . 7 |- ( y = ( card ` z ) -> ( y e. On <-> ( card ` z ) e. On ) )
4	2, 3	mpbiri 248	. . . . . 6 |- ( y = ( card ` z ) -> y e. On )
5	4	adantr 481	. . . . 5 |- ( ( y = ( card ` z ) /\ ( z C_ x /\ A. w e. x E. v e. z w C_ v ) ) -> y e. On )
6	5	exlimiv 1858	. . . 4 |- ( E. z ( y = ( card ` z ) /\ ( z C_ x /\ A. w e. x E. v e. z w C_ v ) ) -> y e. On )
7	6	abssi 3677	. . 3 |- { y | E. z ( y = ( card ` z ) /\ ( z C_ x /\ A. w e. x E. v e. z w C_ v ) ) } C_ On
8		cflem 9068	. . . 4 |- ( x e. On -> E. y E. z ( y = ( card ` z ) /\ ( z C_ x /\ A. w e. x E. v e. z w C_ v ) ) )
9		abn0 3954	. . . 4 |- ( { y | E. z ( y = ( card ` z ) /\ ( z C_ x /\ A. w e. x E. v e. z w C_ v ) ) } =/= (/) <-> E. y E. z ( y = ( card ` z ) /\ ( z C_ x /\ A. w e. x E. v e. z w C_ v ) ) )
10	8, 9	sylibr 224	. . 3 |- ( x e. On -> { y | E. z ( y = ( card ` z ) /\ ( z C_ x /\ A. w e. x E. v e. z w C_ v ) ) } =/= (/) )
11		oninton 7000	. . 3 |- ( ( { y | E. z ( y = ( card ` z ) /\ ( z C_ x /\ A. w e. x E. v e. z w C_ v ) ) } C_ On /\ { y | E. z ( y = ( card ` z ) /\ ( z C_ x /\ A. w e. x E. v e. z w C_ v ) ) } =/= (/) ) -> |^| { y | E. z ( y = ( card ` z ) /\ ( z C_ x /\ A. w e. x E. v e. z w C_ v ) ) } e. On )
12	7, 10, 11	sylancr 695	. 2 |- ( x e. On -> |^| { y | E. z ( y = ( card ` z ) /\ ( z C_ x /\ A. w e. x E. v e. z w C_ v ) ) } e. On )
13	1, 12	fmpti 6383	1 |- cf : On --> On

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   |^|cint 4475   Oncon0 5723   -->wf 5884   ` 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-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-card 8765  df-cf 8767

This theorem is referenced by:  cfub  9071  cardcf  9074  cflecard  9075  cfle  9076  cflim2  9085  cfidm  9097