Theorem cfsmo 9093

Description: The map in cff1 9080 can be assumed to be a strictly monotone ordinal function without loss of generality. (Contributed by Mario Carneiro, 28-Feb-2013.)

Assertion

Ref	Expression
cfsmo	|- ( A e. On -> E. f ( f : ( cf ` A ) --> A /\ Smo f /\ A. z e. A E. w e. ( cf ` A ) z C_ ( f ` w ) ) )

Distinct variable group:   A, f, w, z

Proof of Theorem cfsmo

Dummy variables m h x n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmeq 5324	. . . . 5 |- ( x = z -> dom x = dom z )
2	1	fveq2d 6195	. . . 4 |- ( x = z -> ( h ` dom x ) = ( h ` dom z ) )
3		fveq2 6191	. . . . . . 7 |- ( n = m -> ( x ` n ) = ( x ` m ) )
4		suceq 5790	. . . . . . 7 |- ( ( x ` n ) = ( x ` m ) -> suc ( x ` n ) = suc ( x ` m ) )
5	3, 4	syl 17	. . . . . 6 |- ( n = m -> suc ( x ` n ) = suc ( x ` m ) )
6	5	cbviunv 4559	. . . . 5 |- U_ n e. dom x suc ( x ` n ) = U_ m e. dom x suc ( x ` m )
7		fveq1 6190	. . . . . . 7 |- ( x = z -> ( x ` m ) = ( z ` m ) )
8		suceq 5790	. . . . . . 7 |- ( ( x ` m ) = ( z ` m ) -> suc ( x ` m ) = suc ( z ` m ) )
9	7, 8	syl 17	. . . . . 6 |- ( x = z -> suc ( x ` m ) = suc ( z ` m ) )
10	1, 9	iuneq12d 4546	. . . . 5 |- ( x = z -> U_ m e. dom x suc ( x ` m ) = U_ m e. dom z suc ( z ` m ) )
11	6, 10	syl5eq 2668	. . . 4 |- ( x = z -> U_ n e. dom x suc ( x ` n ) = U_ m e. dom z suc ( z ` m ) )
12	2, 11	uneq12d 3768	. . 3 |- ( x = z -> ( ( h ` dom x ) u. U_ n e. dom x suc ( x ` n ) ) = ( ( h ` dom z ) u. U_ m e. dom z suc ( z ` m ) ) )
13	12	cbvmptv 4750	. 2 |- ( x e. _V |-> ( ( h ` dom x ) u. U_ n e. dom x suc ( x ` n ) ) ) = ( z e. _V |-> ( ( h ` dom z ) u. U_ m e. dom z suc ( z ` m ) ) )
14		eqid 2622	. 2 |- (recs ( ( x e. _V |-> ( ( h ` dom x ) u. U_ n e. dom x suc ( x ` n ) ) ) ) |` ( cf ` A ) ) = (recs ( ( x e. _V |-> ( ( h ` dom x ) u. U_ n e. dom x suc ( x ` n ) ) ) ) |` ( cf ` A ) )
15	13, 14	cfsmolem 9092	1 |- ( A e. On -> E. f ( f : ( cf ` A ) --> A /\ Smo f /\ A. z e. A E. w e. ( cf ` A ) z C_ ( f ` w ) ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   u. cun 3572   C_ wss 3574   U_ciun 4520   |-> cmpt 4729   dom cdm 5114   |` cres 5116   Oncon0 5723   suc csuc 5725   -->wf 5884   ` cfv 5888   Smo wsmo 7442  recscrecs 7467   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:  cfidm  9097  pwcfsdom  9405