Theorem cfflb 9081

Description: If there is a cofinal map from B to A, then B is at least ( cf ` A ). This theorem and cff1 9080 motivate the picture of ( cf ` A ) as the greatest lower bound of the domain of cofinal maps into A. (Contributed by Mario Carneiro, 28-Feb-2013.)

Assertion

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

Distinct variable groups:   A, f, w, z   B, f, w, z

Proof of Theorem cfflb

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

Step	Hyp	Ref	Expression
1		frn 6053	. . . . . . 7 |- ( f : B --> A -> ran f C_ A )
2	1	adantr 481	. . . . . 6 |- ( ( f : B --> A /\ A. z e. A E. w e. B z C_ ( f ` w ) ) -> ran f C_ A )
3		ffn 6045	. . . . . . . . . . . 12 |- ( f : B --> A -> f Fn B )
4		fnfvelrn 6356	. . . . . . . . . . . 12 |- ( ( f Fn B /\ w e. B ) -> ( f ` w ) e. ran f )
5	3, 4	sylan 488	. . . . . . . . . . 11 |- ( ( f : B --> A /\ w e. B ) -> ( f ` w ) e. ran f )
6		sseq2 3627	. . . . . . . . . . . 12 |- ( s = ( f ` w ) -> ( z C_ s <-> z C_ ( f ` w ) ) )
7	6	rspcev 3309	. . . . . . . . . . 11 |- ( ( ( f ` w ) e. ran f /\ z C_ ( f ` w ) ) -> E. s e. ran f z C_ s )
8	5, 7	sylan 488	. . . . . . . . . 10 |- ( ( ( f : B --> A /\ w e. B ) /\ z C_ ( f ` w ) ) -> E. s e. ran f z C_ s )
9	8	exp31 630	. . . . . . . . 9 |- ( f : B --> A -> ( w e. B -> ( z C_ ( f ` w ) -> E. s e. ran f z C_ s ) ) )
10	9	rexlimdv 3030	. . . . . . . 8 |- ( f : B --> A -> ( E. w e. B z C_ ( f ` w ) -> E. s e. ran f z C_ s ) )
11	10	ralimdv 2963	. . . . . . 7 |- ( f : B --> A -> ( A. z e. A E. w e. B z C_ ( f ` w ) -> A. z e. A E. s e. ran f z C_ s ) )
12	11	imp 445	. . . . . 6 |- ( ( f : B --> A /\ A. z e. A E. w e. B z C_ ( f ` w ) ) -> A. z e. A E. s e. ran f z C_ s )
13	2, 12	jca 554	. . . . 5 |- ( ( f : B --> A /\ A. z e. A E. w e. B z C_ ( f ` w ) ) -> ( ran f C_ A /\ A. z e. A E. s e. ran f z C_ s ) )
14		fvex 6201	. . . . . 6 |- ( card ` ran f ) e. _V
15		cfval 9069	. . . . . . . . . . 11 |- ( A e. On -> ( cf ` A ) = |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. s e. y z C_ s ) ) } )
16	15	adantr 481	. . . . . . . . . 10 |- ( ( A e. On /\ B e. On ) -> ( cf ` A ) = |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. s e. y z C_ s ) ) } )
17	16	3ad2ant2 1083	. . . . . . . . 9 |- ( ( x = ( card ` ran f ) /\ ( A e. On /\ B e. On ) /\ ( ran f C_ A /\ A. z e. A E. s e. ran f z C_ s ) ) -> ( cf ` A ) = |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. s e. y z C_ s ) ) } )
18		vex 3203	. . . . . . . . . . . . . 14 |- f e. _V
19	18	rnex 7100	. . . . . . . . . . . . 13 |- ran f e. _V
20		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( y = ran f -> ( card ` y ) = ( card ` ran f ) )
21	20	eqeq2d 2632	. . . . . . . . . . . . . 14 |- ( y = ran f -> ( x = ( card ` y ) <-> x = ( card ` ran f ) ) )
22		sseq1 3626	. . . . . . . . . . . . . . 15 |- ( y = ran f -> ( y C_ A <-> ran f C_ A ) )
23		rexeq 3139	. . . . . . . . . . . . . . . 16 |- ( y = ran f -> ( E. s e. y z C_ s <-> E. s e. ran f z C_ s ) )
24	23	ralbidv 2986	. . . . . . . . . . . . . . 15 |- ( y = ran f -> ( A. z e. A E. s e. y z C_ s <-> A. z e. A E. s e. ran f z C_ s ) )
25	22, 24	anbi12d 747	. . . . . . . . . . . . . 14 |- ( y = ran f -> ( ( y C_ A /\ A. z e. A E. s e. y z C_ s ) <-> ( ran f C_ A /\ A. z e. A E. s e. ran f z C_ s ) ) )
26	21, 25	anbi12d 747	. . . . . . . . . . . . 13 |- ( y = ran f -> ( ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. s e. y z C_ s ) ) <-> ( x = ( card ` ran f ) /\ ( ran f C_ A /\ A. z e. A E. s e. ran f z C_ s ) ) ) )
27	19, 26	spcev 3300	. . . . . . . . . . . 12 |- ( ( x = ( card ` ran f ) /\ ( ran f C_ A /\ A. z e. A E. s e. ran f z C_ s ) ) -> E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. s e. y z C_ s ) ) )
28		abid 2610	. . . . . . . . . . . 12 |- ( x e. { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. s e. y z C_ s ) ) } <-> E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. s e. y z C_ s ) ) )
29	27, 28	sylibr 224	. . . . . . . . . . 11 |- ( ( x = ( card ` ran f ) /\ ( ran f C_ A /\ A. z e. A E. s e. ran f z C_ s ) ) -> x e. { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. s e. y z C_ s ) ) } )
30		intss1 4492	. . . . . . . . . . 11 |- ( x e. { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. s e. y z C_ s ) ) } -> |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. s e. y z C_ s ) ) } C_ x )
31	29, 30	syl 17	. . . . . . . . . 10 |- ( ( x = ( card ` ran f ) /\ ( ran f C_ A /\ A. z e. A E. s e. ran f z C_ s ) ) -> |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. s e. y z C_ s ) ) } C_ x )
32	31	3adant2 1080	. . . . . . . . 9 |- ( ( x = ( card ` ran f ) /\ ( A e. On /\ B e. On ) /\ ( ran f C_ A /\ A. z e. A E. s e. ran f z C_ s ) ) -> |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. s e. y z C_ s ) ) } C_ x )
33	17, 32	eqsstrd 3639	. . . . . . . 8 |- ( ( x = ( card ` ran f ) /\ ( A e. On /\ B e. On ) /\ ( ran f C_ A /\ A. z e. A E. s e. ran f z C_ s ) ) -> ( cf ` A ) C_ x )
34	33	3expib 1268	. . . . . . 7 |- ( x = ( card ` ran f ) -> ( ( ( A e. On /\ B e. On ) /\ ( ran f C_ A /\ A. z e. A E. s e. ran f z C_ s ) ) -> ( cf ` A ) C_ x ) )
35		sseq2 3627	. . . . . . 7 |- ( x = ( card ` ran f ) -> ( ( cf ` A ) C_ x <-> ( cf ` A ) C_ ( card ` ran f ) ) )
36	34, 35	sylibd 229	. . . . . 6 |- ( x = ( card ` ran f ) -> ( ( ( A e. On /\ B e. On ) /\ ( ran f C_ A /\ A. z e. A E. s e. ran f z C_ s ) ) -> ( cf ` A ) C_ ( card ` ran f ) ) )
37	14, 36	vtocle 3282	. . . . 5 |- ( ( ( A e. On /\ B e. On ) /\ ( ran f C_ A /\ A. z e. A E. s e. ran f z C_ s ) ) -> ( cf ` A ) C_ ( card ` ran f ) )
38	13, 37	sylan2 491	. . . 4 |- ( ( ( A e. On /\ B e. On ) /\ ( f : B --> A /\ A. z e. A E. w e. B z C_ ( f ` w ) ) ) -> ( cf ` A ) C_ ( card ` ran f ) )
39		cardidm 8785	. . . . . . 7 |- ( card ` ( card ` ran f ) ) = ( card ` ran f )
40		onss 6990	. . . . . . . . . . . . . 14 |- ( A e. On -> A C_ On )
41	1, 40	sylan9ssr 3617	. . . . . . . . . . . . 13 |- ( ( A e. On /\ f : B --> A ) -> ran f C_ On )
42	41	3adant2 1080	. . . . . . . . . . . 12 |- ( ( A e. On /\ B e. On /\ f : B --> A ) -> ran f C_ On )
43		onssnum 8863	. . . . . . . . . . . 12 |- ( ( ran f e. _V /\ ran f C_ On ) -> ran f e. dom card )
44	19, 42, 43	sylancr 695	. . . . . . . . . . 11 |- ( ( A e. On /\ B e. On /\ f : B --> A ) -> ran f e. dom card )
45		cardid2 8779	. . . . . . . . . . 11 |- ( ran f e. dom card -> ( card ` ran f ) ~~ ran f )
46	44, 45	syl 17	. . . . . . . . . 10 |- ( ( A e. On /\ B e. On /\ f : B --> A ) -> ( card ` ran f ) ~~ ran f )
47		onenon 8775	. . . . . . . . . . . . 13 |- ( B e. On -> B e. dom card )
48		dffn4 6121	. . . . . . . . . . . . . 14 |- ( f Fn B <-> f : B -onto-> ran f )
49	3, 48	sylib 208	. . . . . . . . . . . . 13 |- ( f : B --> A -> f : B -onto-> ran f )
50		fodomnum 8880	. . . . . . . . . . . . 13 |- ( B e. dom card -> ( f : B -onto-> ran f -> ran f ~<_ B ) )
51	47, 49, 50	syl2im 40	. . . . . . . . . . . 12 |- ( B e. On -> ( f : B --> A -> ran f ~<_ B ) )
52	51	imp 445	. . . . . . . . . . 11 |- ( ( B e. On /\ f : B --> A ) -> ran f ~<_ B )
53	52	3adant1 1079	. . . . . . . . . 10 |- ( ( A e. On /\ B e. On /\ f : B --> A ) -> ran f ~<_ B )
54		endomtr 8014	. . . . . . . . . 10 |- ( ( ( card ` ran f ) ~~ ran f /\ ran f ~<_ B ) -> ( card ` ran f ) ~<_ B )
55	46, 53, 54	syl2anc 693	. . . . . . . . 9 |- ( ( A e. On /\ B e. On /\ f : B --> A ) -> ( card ` ran f ) ~<_ B )
56		cardon 8770	. . . . . . . . . . . 12 |- ( card ` ran f ) e. On
57		onenon 8775	. . . . . . . . . . . 12 |- ( ( card ` ran f ) e. On -> ( card ` ran f ) e. dom card )
58	56, 57	ax-mp 5	. . . . . . . . . . 11 |- ( card ` ran f ) e. dom card
59		carddom2 8803	. . . . . . . . . . 11 |- ( ( ( card ` ran f ) e. dom card /\ B e. dom card ) -> ( ( card ` ( card ` ran f ) ) C_ ( card ` B ) <-> ( card ` ran f ) ~<_ B ) )
60	58, 47, 59	sylancr 695	. . . . . . . . . 10 |- ( B e. On -> ( ( card ` ( card ` ran f ) ) C_ ( card ` B ) <-> ( card ` ran f ) ~<_ B ) )
61	60	3ad2ant2 1083	. . . . . . . . 9 |- ( ( A e. On /\ B e. On /\ f : B --> A ) -> ( ( card ` ( card ` ran f ) ) C_ ( card ` B ) <-> ( card ` ran f ) ~<_ B ) )
62	55, 61	mpbird 247	. . . . . . . 8 |- ( ( A e. On /\ B e. On /\ f : B --> A ) -> ( card ` ( card ` ran f ) ) C_ ( card ` B ) )
63		cardonle 8783	. . . . . . . . 9 |- ( B e. On -> ( card ` B ) C_ B )
64	63	3ad2ant2 1083	. . . . . . . 8 |- ( ( A e. On /\ B e. On /\ f : B --> A ) -> ( card ` B ) C_ B )
65	62, 64	sstrd 3613	. . . . . . 7 |- ( ( A e. On /\ B e. On /\ f : B --> A ) -> ( card ` ( card ` ran f ) ) C_ B )
66	39, 65	syl5eqssr 3650	. . . . . 6 |- ( ( A e. On /\ B e. On /\ f : B --> A ) -> ( card ` ran f ) C_ B )
67	66	3expa 1265	. . . . 5 |- ( ( ( A e. On /\ B e. On ) /\ f : B --> A ) -> ( card ` ran f ) C_ B )
68	67	adantrr 753	. . . 4 |- ( ( ( A e. On /\ B e. On ) /\ ( f : B --> A /\ A. z e. A E. w e. B z C_ ( f ` w ) ) ) -> ( card ` ran f ) C_ B )
69	38, 68	sstrd 3613	. . 3 |- ( ( ( A e. On /\ B e. On ) /\ ( f : B --> A /\ A. z e. A E. w e. B z C_ ( f ` w ) ) ) -> ( cf ` A ) C_ B )
70	69	ex 450	. 2 |- ( ( A e. On /\ B e. On ) -> ( ( f : B --> A /\ A. z e. A E. w e. B z C_ ( f ` w ) ) -> ( cf ` A ) C_ B ) )
71	70	exlimdv 1861	1 |- ( ( A e. On /\ B e. On ) -> ( E. f ( f : B --> A /\ A. z e. A E. w e. B z C_ ( f ` w ) ) -> ( cf ` A ) C_ B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   |^|cint 4475   class class class wbr 4653   dom cdm 5114   ran crn 5115   Oncon0 5723   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888   ~~ cen 7952   ~<_ cdom 7953   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-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-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:  cfsmolem  9092  cfcoflem  9094  cfcof  9096  inar1  9597