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Theorem cfflb 7901
Description: If there is a cofinal map from  B to  A, then  B is at least  ( cf `  A
). This theorem and cff1 7900 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 5411 . . . . . . 7  |-  ( f : B --> A  ->  ran  f  C_  A )
21adantr 451 . . . . . 6  |-  ( ( f : B --> A  /\  A. z  e.  A  E. w  e.  B  z  C_  ( f `  w
) )  ->  ran  f  C_  A )
3 ffn 5405 . . . . . . . . . . . 12  |-  ( f : B --> A  -> 
f  Fn  B )
4 fnfvelrn 5678 . . . . . . . . . . . 12  |-  ( ( f  Fn  B  /\  w  e.  B )  ->  ( f `  w
)  e.  ran  f
)
53, 4sylan 457 . . . . . . . . . . 11  |-  ( ( f : B --> A  /\  w  e.  B )  ->  ( f `  w
)  e.  ran  f
)
6 sseq2 3213 . . . . . . . . . . . 12  |-  ( s  =  ( f `  w )  ->  (
z  C_  s  <->  z  C_  ( f `  w
) ) )
76rspcev 2897 . . . . . . . . . . 11  |-  ( ( ( f `  w
)  e.  ran  f  /\  z  C_  ( f `
 w ) )  ->  E. s  e.  ran  f  z  C_  s )
85, 7sylan 457 . . . . . . . . . 10  |-  ( ( ( f : B --> A  /\  w  e.  B
)  /\  z  C_  ( f `  w
) )  ->  E. s  e.  ran  f  z  C_  s )
98exp31 587 . . . . . . . . 9  |-  ( f : B --> A  -> 
( w  e.  B  ->  ( z  C_  (
f `  w )  ->  E. s  e.  ran  f  z  C_  s ) ) )
109rexlimdv 2679 . . . . . . . 8  |-  ( f : B --> A  -> 
( E. w  e.  B  z  C_  (
f `  w )  ->  E. s  e.  ran  f  z  C_  s ) )
1110ralimdv 2635 . . . . . . 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 418 . . . . . 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 518 . . . . 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 5555 . . . . . 6  |-  ( card `  ran  f )  e. 
_V
15 cfval 7889 . . . . . . . . . . 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 451 . . . . . . . . . 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 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 ) )  -> 
( cf `  A
)  =  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. s  e.  y  z  C_  s ) ) } )
18 vex 2804 . . . . . . . . . . . . . 14  |-  f  e. 
_V
1918rnex 4958 . . . . . . . . . . . . 13  |-  ran  f  e.  _V
20 fveq2 5541 . . . . . . . . . . . . . . 15  |-  ( y  =  ran  f  -> 
( card `  y )  =  ( card `  ran  f ) )
2120eqeq2d 2307 . . . . . . . . . . . . . 14  |-  ( y  =  ran  f  -> 
( x  =  (
card `  y )  <->  x  =  ( card `  ran  f ) ) )
22 sseq1 3212 . . . . . . . . . . . . . . 15  |-  ( y  =  ran  f  -> 
( y  C_  A  <->  ran  f  C_  A )
)
23 rexeq 2750 . . . . . . . . . . . . . . . 16  |-  ( y  =  ran  f  -> 
( E. s  e.  y  z  C_  s  <->  E. s  e.  ran  f 
z  C_  s )
)
2423ralbidv 2576 . . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . 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 2888 . . . . . . . . . . . 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 2284 . . . . . . . . . . . 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 203 . . . . . . . . . . 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 3893 . . . . . . . . . . 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 15 . . . . . . . . . 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 974 . . . . . . . . 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 3225 . . . . . . . 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 1154 . . . . . . 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 3213 . . . . . . 7  |-  ( x  =  ( card `  ran  f )  ->  (
( cf `  A
)  C_  x  <->  ( cf `  A )  C_  ( card `  ran  f ) ) )
3634, 35sylibd 205 . . . . . 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 2870 . . . . 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 460 . . . 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 7608 . . . . . . 7  |-  ( card `  ( card `  ran  f ) )  =  ( card `  ran  f )
40 onss 4598 . . . . . . . . . . . . . 14  |-  ( A  e.  On  ->  A  C_  On )
411, 40sylan9ssr 3206 . . . . . . . . . . . . 13  |-  ( ( A  e.  On  /\  f : B --> A )  ->  ran  f  C_  On )
42413adant2 974 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  ->  ran  f  C_  On )
43 onssnum 7683 . . . . . . . . . . . 12  |-  ( ( ran  f  e.  _V  /\ 
ran  f  C_  On )  ->  ran  f  e.  dom  card )
4419, 42, 43sylancr 644 . . . . . . . . . . 11  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  ->  ran  f  e.  dom  card )
45 cardid2 7602 . . . . . . . . . . 11  |-  ( ran  f  e.  dom  card  -> 
( card `  ran  f ) 
~~  ran  f )
4644, 45syl 15 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  -> 
( card `  ran  f ) 
~~  ran  f )
47 onenon 7598 . . . . . . . . . . . . 13  |-  ( B  e.  On  ->  B  e.  dom  card )
48 dffn4 5473 . . . . . . . . . . . . . 14  |-  ( f  Fn  B  <->  f : B -onto-> ran  f )
493, 48sylib 188 . . . . . . . . . . . . 13  |-  ( f : B --> A  -> 
f : B -onto-> ran  f )
50 fodomnum 7700 . . . . . . . . . . . . 13  |-  ( B  e.  dom  card  ->  ( f : B -onto-> ran  f  ->  ran  f  ~<_  B ) )
5147, 49, 50syl2im 34 . . . . . . . . . . . 12  |-  ( B  e.  On  ->  (
f : B --> A  ->  ran  f  ~<_  B )
)
5251imp 418 . . . . . . . . . . 11  |-  ( ( B  e.  On  /\  f : B --> A )  ->  ran  f  ~<_  B )
53523adant1 973 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  ->  ran  f  ~<_  B )
54 endomtr 6935 . . . . . . . . . 10  |-  ( ( ( card `  ran  f )  ~~  ran  f  /\  ran  f  ~<_  B )  ->  ( card ` 
ran  f )  ~<_  B )
5546, 53, 54syl2anc 642 . . . . . . . . 9  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  -> 
( card `  ran  f )  ~<_  B )
56 cardon 7593 . . . . . . . . . . . 12  |-  ( card `  ran  f )  e.  On
57 onenon 7598 . . . . . . . . . . . 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 7626 . . . . . . . . . . 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 644 . . . . . . . . . 10  |-  ( B  e.  On  ->  (
( card `  ( card ` 
ran  f ) ) 
C_  ( card `  B
)  <->  ( card `  ran  f )  ~<_  B ) )
61603ad2ant2 977 . . . . . . . . 9  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  -> 
( ( card `  ( card `  ran  f ) )  C_  ( card `  B )  <->  ( card ` 
ran  f )  ~<_  B ) )
6255, 61mpbird 223 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  -> 
( card `  ( card ` 
ran  f ) ) 
C_  ( card `  B
) )
63 cardonle 7606 . . . . . . . . 9  |-  ( B  e.  On  ->  ( card `  B )  C_  B )
64633ad2ant2 977 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  -> 
( card `  B )  C_  B )
6562, 64sstrd 3202 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  -> 
( card `  ( card ` 
ran  f ) ) 
C_  B )
6639, 65syl5eqssr 3236 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On  /\  f : B --> A )  -> 
( card `  ran  f ) 
C_  B )
67663expa 1151 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  f : B --> A )  ->  ( card `  ran  f ) 
C_  B )
6867adantrr 697 . . . 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 3202 . . 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 423 . 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 1626 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 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165   |^|cint 3878   class class class wbr 4039   Oncon0 4408   dom cdm 4705   ran crn 4706    Fn wfn 5266   -->wf 5267   -onto->wfo 5269   ` cfv 5271    ~~ cen 6876    ~<_ cdom 6877   cardccrd 7584   cfccf 7586
This theorem is referenced by:  cfsmolem  7912  cfcoflem  7914  cfcof  7916  inar1  8413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-card 7588  df-cf 7590  df-acn 7591
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