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Theorem cfflb 8095
Description: If there is a cofinal map from  B to  A, then  B is at least  ( cf `  A
). This theorem and cff1 8094 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 5556 . . . . . . 7  |-  ( f : B --> A  ->  ran  f  C_  A )
21adantr 452 . . . . . 6  |-  ( ( f : B --> A  /\  A. z  e.  A  E. w  e.  B  z  C_  ( f `  w
) )  ->  ran  f  C_  A )
3 ffn 5550 . . . . . . . . . . . 12  |-  ( f : B --> A  -> 
f  Fn  B )
4 fnfvelrn 5826 . . . . . . . . . . . 12  |-  ( ( f  Fn  B  /\  w  e.  B )  ->  ( f `  w
)  e.  ran  f
)
53, 4sylan 458 . . . . . . . . . . 11  |-  ( ( f : B --> A  /\  w  e.  B )  ->  ( f `  w
)  e.  ran  f
)
6 sseq2 3330 . . . . . . . . . . . 12  |-  ( s  =  ( f `  w )  ->  (
z  C_  s  <->  z  C_  ( f `  w
) ) )
76rspcev 3012 . . . . . . . . . . 11  |-  ( ( ( f `  w
)  e.  ran  f  /\  z  C_  ( f `
 w ) )  ->  E. s  e.  ran  f  z  C_  s )
85, 7sylan 458 . . . . . . . . . 10  |-  ( ( ( f : B --> A  /\  w  e.  B
)  /\  z  C_  ( f `  w
) )  ->  E. s  e.  ran  f  z  C_  s )
98exp31 588 . . . . . . . . 9  |-  ( f : B --> A  -> 
( w  e.  B  ->  ( z  C_  (
f `  w )  ->  E. s  e.  ran  f  z  C_  s ) ) )
109rexlimdv 2789 . . . . . . . 8  |-  ( f : B --> A  -> 
( E. w  e.  B  z  C_  (
f `  w )  ->  E. s  e.  ran  f  z  C_  s ) )
1110ralimdv 2745 . . . . . . 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 419 . . . . . 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 519 . . . . 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 5701 . . . . . 6  |-  ( card `  ran  f )  e. 
_V
15 cfval 8083 . . . . . . . . . . 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 452 . . . . . . . . . 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 979 . . . . . . . . 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 2919 . . . . . . . . . . . . . 14  |-  f  e. 
_V
1918rnex 5092 . . . . . . . . . . . . 13  |-  ran  f  e.  _V
20 fveq2 5687 . . . . . . . . . . . . . . 15  |-  ( y  =  ran  f  -> 
( card `  y )  =  ( card `  ran  f ) )
2120eqeq2d 2415 . . . . . . . . . . . . . 14  |-  ( y  =  ran  f  -> 
( x  =  (
card `  y )  <->  x  =  ( card `  ran  f ) ) )
22 sseq1 3329 . . . . . . . . . . . . . . 15  |-  ( y  =  ran  f  -> 
( y  C_  A  <->  ran  f  C_  A )
)
23 rexeq 2865 . . . . . . . . . . . . . . . 16  |-  ( y  =  ran  f  -> 
( E. s  e.  y  z  C_  s  <->  E. s  e.  ran  f 
z  C_  s )
)
2423ralbidv 2686 . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . 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 3003 . . . . . . . . . . . 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 2392 . . . . . . . . . . . 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 204 . . . . . . . . . . 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 4025 . . . . . . . . . . 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 976 . . . . . . . . 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 3342 . . . . . . . 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 1156 . . . . . . 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 3330 . . . . . . 7  |-  ( x  =  ( card `  ran  f )  ->  (
( cf `  A
)  C_  x  <->  ( cf `  A )  C_  ( card `  ran  f ) ) )
3634, 35sylibd 206 . . . . . 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 2985 . . . . 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 461 . . . 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 7802 . . . . . . 7  |-  ( card `  ( card `  ran  f ) )  =  ( card `  ran  f )
40 onss 4730 . . . . . . . . . . . . . 14  |-  ( A  e.  On  ->  A  C_  On )
411, 40sylan9ssr 3322 . . . . . . . . . . . . 13  |-  ( ( A  e.  On  /\  f : B --> A )  ->  ran  f  C_  On )
42413adant2 976 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  ->  ran  f  C_  On )
43 onssnum 7877 . . . . . . . . . . . 12  |-  ( ( ran  f  e.  _V  /\ 
ran  f  C_  On )  ->  ran  f  e.  dom  card )
4419, 42, 43sylancr 645 . . . . . . . . . . 11  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  ->  ran  f  e.  dom  card )
45 cardid2 7796 . . . . . . . . . . 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 7792 . . . . . . . . . . . . 13  |-  ( B  e.  On  ->  B  e.  dom  card )
48 dffn4 5618 . . . . . . . . . . . . . 14  |-  ( f  Fn  B  <->  f : B -onto-> ran  f )
493, 48sylib 189 . . . . . . . . . . . . 13  |-  ( f : B --> A  -> 
f : B -onto-> ran  f )
50 fodomnum 7894 . . . . . . . . . . . . 13  |-  ( B  e.  dom  card  ->  ( f : B -onto-> ran  f  ->  ran  f  ~<_  B ) )
5147, 49, 50syl2im 36 . . . . . . . . . . . 12  |-  ( B  e.  On  ->  (
f : B --> A  ->  ran  f  ~<_  B )
)
5251imp 419 . . . . . . . . . . 11  |-  ( ( B  e.  On  /\  f : B --> A )  ->  ran  f  ~<_  B )
53523adant1 975 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  ->  ran  f  ~<_  B )
54 endomtr 7124 . . . . . . . . . 10  |-  ( ( ( card `  ran  f )  ~~  ran  f  /\  ran  f  ~<_  B )  ->  ( card ` 
ran  f )  ~<_  B )
5546, 53, 54syl2anc 643 . . . . . . . . 9  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  -> 
( card `  ran  f )  ~<_  B )
56 cardon 7787 . . . . . . . . . . . 12  |-  ( card `  ran  f )  e.  On
57 onenon 7792 . . . . . . . . . . . 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 7820 . . . . . . . . . . 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 645 . . . . . . . . . 10  |-  ( B  e.  On  ->  (
( card `  ( card ` 
ran  f ) ) 
C_  ( card `  B
)  <->  ( card `  ran  f )  ~<_  B ) )
61603ad2ant2 979 . . . . . . . . 9  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  -> 
( ( card `  ( card `  ran  f ) )  C_  ( card `  B )  <->  ( card ` 
ran  f )  ~<_  B ) )
6255, 61mpbird 224 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  -> 
( card `  ( card ` 
ran  f ) ) 
C_  ( card `  B
) )
63 cardonle 7800 . . . . . . . . 9  |-  ( B  e.  On  ->  ( card `  B )  C_  B )
64633ad2ant2 979 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  -> 
( card `  B )  C_  B )
6562, 64sstrd 3318 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  -> 
( card `  ( card ` 
ran  f ) ) 
C_  B )
6639, 65syl5eqssr 3353 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  -> 
( card `  ran  f ) 
C_  B )
67663expa 1153 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  f : B --> A )  ->  ( card `  ran  f ) 
C_  B )
6867adantrr 698 . . . 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 3318 . . 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 424 . 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 1643 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 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   {cab 2390   A.wral 2666   E.wrex 2667   _Vcvv 2916    C_ wss 3280   |^|cint 4010   class class class wbr 4172   Oncon0 4541   dom cdm 4837   ran crn 4838    Fn wfn 5408   -->wf 5409   -onto->wfo 5411   ` cfv 5413    ~~ cen 7065    ~<_ cdom 7066   cardccrd 7778   cfccf 7780
This theorem is referenced by:  cfsmolem  8106  cfcoflem  8108  cfcof  8110  inar1  8606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-card 7782  df-cf 7784  df-acn 7785
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