Theorem cfidm 9097

Description: The cofinality function is idempotent. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)

Assertion

Ref	Expression
cfidm	|- ( cf ` ( cf ` A ) ) = ( cf ` A )

Proof of Theorem cfidm

Dummy variables f x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cfle 9076	. . . 4 |- ( cf ` ( cf ` A ) ) C_ ( cf ` A )
2	1	a1i 11	. . 3 |- ( A e. On -> ( cf ` ( cf ` A ) ) C_ ( cf ` A ) )
3		cfsmo 9093	. . . 4 |- ( A e. On -> E. f ( f : ( cf ` A ) --> A /\ Smo f /\ A. x e. A E. y e. ( cf ` A ) x C_ ( f ` y ) ) )
4		cfon 9077	. . . . 5 |- ( cf ` A ) e. On
5		cfcoflem 9094	. . . . 5 |- ( ( A e. On /\ ( cf ` A ) e. On ) -> ( E. f ( f : ( cf ` A ) --> A /\ Smo f /\ A. x e. A E. y e. ( cf ` A ) x C_ ( f ` y ) ) -> ( cf ` A ) C_ ( cf ` ( cf ` A ) ) ) )
6	4, 5	mpan2 707	. . . 4 |- ( A e. On -> ( E. f ( f : ( cf ` A ) --> A /\ Smo f /\ A. x e. A E. y e. ( cf ` A ) x C_ ( f ` y ) ) -> ( cf ` A ) C_ ( cf ` ( cf ` A ) ) ) )
7	3, 6	mpd 15	. . 3 |- ( A e. On -> ( cf ` A ) C_ ( cf ` ( cf ` A ) ) )
8	2, 7	eqssd 3620	. 2 |- ( A e. On -> ( cf ` ( cf ` A ) ) = ( cf ` A ) )
9		cf0 9073	. . 3 |- ( cf ` (/) ) = (/)
10		cff 9070	. . . . . . 7 |- cf : On --> On
11	10	fdmi 6052	. . . . . 6 |- dom cf = On
12	11	eleq2i 2693	. . . . 5 |- ( A e. dom cf <-> A e. On )
13		ndmfv 6218	. . . . 5 |- ( -. A e. dom cf -> ( cf ` A ) = (/) )
14	12, 13	sylnbir 321	. . . 4 |- ( -. A e. On -> ( cf ` A ) = (/) )
15	14	fveq2d 6195	. . 3 |- ( -. A e. On -> ( cf ` ( cf ` A ) ) = ( cf ` (/) ) )
16	9, 15, 14	3eqtr4a 2682	. 2 |- ( -. A e. On -> ( cf ` ( cf ` A ) ) = ( cf ` A ) )
17	8, 16	pm2.61i 176	1 |- ( cf ` ( cf ` A ) ) = ( cf ` A )

Syntax hints:   -. wn 3   -> wi 4   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   dom cdm 5114   Oncon0 5723   -->wf 5884   ` cfv 5888   Smo wsmo 7442   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-rep 4771  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-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-se 5074  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-pred 5680  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-smo 7443  df-recs 7468  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-card 8765  df-cf 8767  df-acn 8768

This theorem is referenced by: (None)