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Theorem isinffi 7625
Description: An infinite set contains subsets equinumerous to every finite set. Extension of isinf 7076 from finite ordinals to all finite sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
Assertion
Ref Expression
isinffi  |-  ( ( -.  A  e.  Fin  /\  B  e.  Fin )  ->  E. f  f : B -1-1-> A )
Distinct variable groups:    A, f    B, f

Proof of Theorem isinffi
Dummy variables  c 
a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ficardom 7594 . . 3  |-  ( B  e.  Fin  ->  ( card `  B )  e. 
om )
2 isinf 7076 . . 3  |-  ( -.  A  e.  Fin  ->  A. a  e.  om  E. c ( c  C_  A  /\  c  ~~  a
) )
3 breq2 4027 . . . . . 6  |-  ( a  =  ( card `  B
)  ->  ( c  ~~  a  <->  c  ~~  ( card `  B ) ) )
43anbi2d 684 . . . . 5  |-  ( a  =  ( card `  B
)  ->  ( (
c  C_  A  /\  c  ~~  a )  <->  ( c  C_  A  /\  c  ~~  ( card `  B )
) ) )
54exbidv 1612 . . . 4  |-  ( a  =  ( card `  B
)  ->  ( E. c ( c  C_  A  /\  c  ~~  a
)  <->  E. c ( c 
C_  A  /\  c  ~~  ( card `  B
) ) ) )
65rspcva 2882 . . 3  |-  ( ( ( card `  B
)  e.  om  /\  A. a  e.  om  E. c ( c  C_  A  /\  c  ~~  a
) )  ->  E. c
( c  C_  A  /\  c  ~~  ( card `  B ) ) )
71, 2, 6syl2anr 464 . 2  |-  ( ( -.  A  e.  Fin  /\  B  e.  Fin )  ->  E. c ( c 
C_  A  /\  c  ~~  ( card `  B
) ) )
8 simprr 733 . . . . . . . 8  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  -> 
c  ~~  ( card `  B ) )
9 ficardid 7595 . . . . . . . . 9  |-  ( B  e.  Fin  ->  ( card `  B )  ~~  B )
109ad2antlr 707 . . . . . . . 8  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  -> 
( card `  B )  ~~  B )
11 entr 6913 . . . . . . . 8  |-  ( ( c  ~~  ( card `  B )  /\  ( card `  B )  ~~  B )  ->  c  ~~  B )
128, 10, 11syl2anc 642 . . . . . . 7  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  -> 
c  ~~  B )
13 ensym 6910 . . . . . . 7  |-  ( c 
~~  B  ->  B  ~~  c )
1412, 13syl 15 . . . . . 6  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  ->  B  ~~  c )
15 bren 6871 . . . . . 6  |-  ( B 
~~  c  <->  E. f 
f : B -1-1-onto-> c )
1614, 15sylib 188 . . . . 5  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  ->  E. f  f : B
-1-1-onto-> c )
17 f1of1 5471 . . . . . . . . 9  |-  ( f : B -1-1-onto-> c  ->  f : B -1-1-> c )
1817adantl 452 . . . . . . . 8  |-  ( ( ( ( -.  A  e.  Fin  /\  B  e. 
Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  /\  f : B -1-1-onto-> c )  ->  f : B -1-1-> c )
19 simplrl 736 . . . . . . . 8  |-  ( ( ( ( -.  A  e.  Fin  /\  B  e. 
Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  /\  f : B -1-1-onto-> c )  ->  c  C_  A )
20 f1ss 5442 . . . . . . . 8  |-  ( ( f : B -1-1-> c  /\  c  C_  A
)  ->  f : B -1-1-> A )
2118, 19, 20syl2anc 642 . . . . . . 7  |-  ( ( ( ( -.  A  e.  Fin  /\  B  e. 
Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  /\  f : B -1-1-onto-> c )  ->  f : B -1-1-> A )
2221ex 423 . . . . . 6  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  -> 
( f : B -1-1-onto-> c  ->  f : B -1-1-> A
) )
2322eximdv 1608 . . . . 5  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  -> 
( E. f  f : B -1-1-onto-> c  ->  E. f 
f : B -1-1-> A
) )
2416, 23mpd 14 . . . 4  |-  ( ( ( -.  A  e. 
Fin  /\  B  e.  Fin )  /\  (
c  C_  A  /\  c  ~~  ( card `  B
) ) )  ->  E. f  f : B -1-1-> A )
2524ex 423 . . 3  |-  ( ( -.  A  e.  Fin  /\  B  e.  Fin )  ->  ( ( c  C_  A  /\  c  ~~  ( card `  B ) )  ->  E. f  f : B -1-1-> A ) )
2625exlimdv 1664 . 2  |-  ( ( -.  A  e.  Fin  /\  B  e.  Fin )  ->  ( E. c ( c  C_  A  /\  c  ~~  ( card `  B
) )  ->  E. f 
f : B -1-1-> A
) )
277, 26mpd 14 1  |-  ( ( -.  A  e.  Fin  /\  B  e.  Fin )  ->  E. f  f : B -1-1-> A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   class class class wbr 4023   omcom 4656   -1-1->wf1 5252   -1-1-onto->wf1o 5254   ` cfv 5255    ~~ cen 6860   Fincfn 6863   cardccrd 7568
This theorem is referenced by:  fidomtri  7626  hashdom  11361  erdsze2lem1  23734  eldioph2lem2  26840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572
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