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Theorem cfflb 8141
Description: If there is a cofinal map from  B to  A, then  B is at least  ( cf `  A
). This theorem and cff1 8140 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.
StepHypRef Expression
1 frn 5599 . . . . . . 7  |-  ( f : B --> A  ->  ran  f  C_  A )
21adantr 453 . . . . . 6  |-  ( ( f : B --> A  /\  A. z  e.  A  E. w  e.  B  z  C_  ( f `  w
) )  ->  ran  f  C_  A )
3 ffn 5593 . . . . . . . . . . . 12  |-  ( f : B --> A  -> 
f  Fn  B )
4 fnfvelrn 5869 . . . . . . . . . . . 12  |-  ( ( f  Fn  B  /\  w  e.  B )  ->  ( f `  w
)  e.  ran  f
)
53, 4sylan 459 . . . . . . . . . . 11  |-  ( ( f : B --> A  /\  w  e.  B )  ->  ( f `  w
)  e.  ran  f
)
6 sseq2 3372 . . . . . . . . . . . 12  |-  ( s  =  ( f `  w )  ->  (
z  C_  s  <->  z  C_  ( f `  w
) ) )
76rspcev 3054 . . . . . . . . . . 11  |-  ( ( ( f `  w
)  e.  ran  f  /\  z  C_  ( f `
 w ) )  ->  E. s  e.  ran  f  z  C_  s )
85, 7sylan 459 . . . . . . . . . 10  |-  ( ( ( f : B --> A  /\  w  e.  B
)  /\  z  C_  ( f `  w
) )  ->  E. s  e.  ran  f  z  C_  s )
98exp31 589 . . . . . . . . 9  |-  ( f : B --> A  -> 
( w  e.  B  ->  ( z  C_  (
f `  w )  ->  E. s  e.  ran  f  z  C_  s ) ) )
109rexlimdv 2831 . . . . . . . 8  |-  ( f : B --> A  -> 
( E. w  e.  B  z  C_  (
f `  w )  ->  E. s  e.  ran  f  z  C_  s ) )
1110ralimdv 2787 . . . . . . 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 ) )
1211imp 420 . . . . . 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 )
132, 12jca 520 . . . . 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 5744 . . . . . 6  |-  ( card `  ran  f )  e. 
_V
15 cfval 8129 . . . . . . . . . . 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 ) ) } )
1615adantr 453 . . . . . . . . . 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 ) ) } )
17163ad2ant2 980 . . . . . . . . 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 2961 . . . . . . . . . . . . . 14  |-  f  e. 
_V
1918rnex 5135 . . . . . . . . . . . . 13  |-  ran  f  e.  _V
20 fveq2 5730 . . . . . . . . . . . . . . 15  |-  ( y  =  ran  f  -> 
( card `  y )  =  ( card `  ran  f ) )
2120eqeq2d 2449 . . . . . . . . . . . . . 14  |-  ( y  =  ran  f  -> 
( x  =  (
card `  y )  <->  x  =  ( card `  ran  f ) ) )
22 sseq1 3371 . . . . . . . . . . . . . . 15  |-  ( y  =  ran  f  -> 
( y  C_  A  <->  ran  f  C_  A )
)
23 rexeq 2907 . . . . . . . . . . . . . . . 16  |-  ( y  =  ran  f  -> 
( E. s  e.  y  z  C_  s  <->  E. s  e.  ran  f 
z  C_  s )
)
2423ralbidv 2727 . . . . . . . . . . . . . . 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 ) )
2522, 24anbi12d 693 . . . . . . . . . . . . . 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 ) ) )
2621, 25anbi12d 693 . . . . . . . . . . . . 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 ) ) ) )
2719, 26spcev 3045 . . . . . . . . . . . 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 2426 . . . . . . . . . . . 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 ) ) )
2927, 28sylibr 205 . . . . . . . . . . 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 4067 . . . . . . . . . . 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 )
3129, 30syl 16 . . . . . . . . . 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 )
32313adant2 977 . . . . . . . . 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 )
3317, 32eqsstrd 3384 . . . . . . . 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 )
34333expib 1157 . . . . . . 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 3372 . . . . . . 7  |-  ( x  =  ( card `  ran  f )  ->  (
( cf `  A
)  C_  x  <->  ( cf `  A )  C_  ( card `  ran  f ) ) )
3634, 35sylibd 207 . . . . . 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 ) ) )
3714, 36vtocle 3027 . . . . 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 ) )
3813, 37sylan2 462 . . . 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 7848 . . . . . . 7  |-  ( card `  ( card `  ran  f ) )  =  ( card `  ran  f )
40 onss 4773 . . . . . . . . . . . . . 14  |-  ( A  e.  On  ->  A  C_  On )
411, 40sylan9ssr 3364 . . . . . . . . . . . . 13  |-  ( ( A  e.  On  /\  f : B --> A )  ->  ran  f  C_  On )
42413adant2 977 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  ->  ran  f  C_  On )
43 onssnum 7923 . . . . . . . . . . . 12  |-  ( ( ran  f  e.  _V  /\ 
ran  f  C_  On )  ->  ran  f  e.  dom  card )
4419, 42, 43sylancr 646 . . . . . . . . . . 11  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  ->  ran  f  e.  dom  card )
45 cardid2 7842 . . . . . . . . . . 11  |-  ( ran  f  e.  dom  card  -> 
( card `  ran  f ) 
~~  ran  f )
4644, 45syl 16 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  -> 
( card `  ran  f ) 
~~  ran  f )
47 onenon 7838 . . . . . . . . . . . . 13  |-  ( B  e.  On  ->  B  e.  dom  card )
48 dffn4 5661 . . . . . . . . . . . . . 14  |-  ( f  Fn  B  <->  f : B -onto-> ran  f )
493, 48sylib 190 . . . . . . . . . . . . 13  |-  ( f : B --> A  -> 
f : B -onto-> ran  f )
50 fodomnum 7940 . . . . . . . . . . . . 13  |-  ( B  e.  dom  card  ->  ( f : B -onto-> ran  f  ->  ran  f  ~<_  B ) )
5147, 49, 50syl2im 37 . . . . . . . . . . . 12  |-  ( B  e.  On  ->  (
f : B --> A  ->  ran  f  ~<_  B )
)
5251imp 420 . . . . . . . . . . 11  |-  ( ( B  e.  On  /\  f : B --> A )  ->  ran  f  ~<_  B )
53523adant1 976 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  ->  ran  f  ~<_  B )
54 endomtr 7167 . . . . . . . . . 10  |-  ( ( ( card `  ran  f )  ~~  ran  f  /\  ran  f  ~<_  B )  ->  ( card ` 
ran  f )  ~<_  B )
5546, 53, 54syl2anc 644 . . . . . . . . 9  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  -> 
( card `  ran  f )  ~<_  B )
56 cardon 7833 . . . . . . . . . . . 12  |-  ( card `  ran  f )  e.  On
57 onenon 7838 . . . . . . . . . . . 12  |-  ( (
card `  ran  f )  e.  On  ->  ( card `  ran  f )  e.  dom  card )
5856, 57ax-mp 8 . . . . . . . . . . 11  |-  ( card `  ran  f )  e. 
dom  card
59 carddom2 7866 . . . . . . . . . . 11  |-  ( ( ( card `  ran  f )  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  ( card `  ran  f ) )  C_  ( card `  B )  <->  (
card `  ran  f )  ~<_  B ) )
6058, 47, 59sylancr 646 . . . . . . . . . 10  |-  ( B  e.  On  ->  (
( card `  ( card ` 
ran  f ) ) 
C_  ( card `  B
)  <->  ( card `  ran  f )  ~<_  B ) )
61603ad2ant2 980 . . . . . . . . 9  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  -> 
( ( card `  ( card `  ran  f ) )  C_  ( card `  B )  <->  ( card ` 
ran  f )  ~<_  B ) )
6255, 61mpbird 225 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  -> 
( card `  ( card ` 
ran  f ) ) 
C_  ( card `  B
) )
63 cardonle 7846 . . . . . . . . 9  |-  ( B  e.  On  ->  ( card `  B )  C_  B )
64633ad2ant2 980 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  -> 
( card `  B )  C_  B )
6562, 64sstrd 3360 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  -> 
( card `  ( card ` 
ran  f ) ) 
C_  B )
6639, 65syl5eqssr 3395 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  -> 
( card `  ran  f ) 
C_  B )
67663expa 1154 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  f : B --> A )  ->  ( card `  ran  f ) 
C_  B )
6867adantrr 699 . . . 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 )
6938, 68sstrd 3360 . . 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 )
7069ex 425 . 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 ) )
7170exlimdv 1647 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 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   E.wrex 2708   _Vcvv 2958    C_ wss 3322   |^|cint 4052   class class class wbr 4214   Oncon0 4583   dom cdm 4880   ran crn 4881    Fn wfn 5451   -->wf 5452   -onto->wfo 5454   ` cfv 5456    ~~ cen 7108    ~<_ cdom 7109   cardccrd 7824   cfccf 7826
This theorem is referenced by:  cfsmolem  8152  cfcoflem  8154  cfcof  8156  inar1  8652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-suc 4589  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-card 7828  df-cf 7830  df-acn 7831
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