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Theorem fodomfi2 7941
Description: Onto functions define dominance when a finite number of choices need to be made. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Assertion
Ref Expression
fodomfi2  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  B  ~<_  A )

Proof of Theorem fodomfi2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fofn 5655 . . . 4  |-  ( F : A -onto-> B  ->  F  Fn  A )
213ad2ant3 980 . . 3  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  F  Fn  A )
3 forn 5656 . . . . 5  |-  ( F : A -onto-> B  ->  ran  F  =  B )
4 eqimss2 3401 . . . . 5  |-  ( ran 
F  =  B  ->  B  C_  ran  F )
53, 4syl 16 . . . 4  |-  ( F : A -onto-> B  ->  B  C_  ran  F )
653ad2ant3 980 . . 3  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  B  C_ 
ran  F )
7 simp2 958 . . 3  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  B  e.  Fin )
8 fipreima 7412 . . 3  |-  ( ( F  Fn  A  /\  B  C_  ran  F  /\  B  e.  Fin )  ->  E. x  e.  ( ~P A  i^i  Fin ) ( F "
x )  =  B )
92, 6, 7, 8syl3anc 1184 . 2  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  E. x  e.  ( ~P A  i^i  Fin ) ( F "
x )  =  B )
10 inss2 3562 . . . . . . . . 9  |-  ( ~P A  i^i  Fin )  C_ 
Fin
1110sseli 3344 . . . . . . . 8  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  Fin )
1211adantl 453 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  Fin )
13 finnum 7835 . . . . . . 7  |-  ( x  e.  Fin  ->  x  e.  dom  card )
1412, 13syl 16 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  dom  card )
15 simpl3 962 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  F : A -onto-> B )
16 fofun 5654 . . . . . . . 8  |-  ( F : A -onto-> B  ->  Fun  F )
1715, 16syl 16 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  Fun  F )
18 inss1 3561 . . . . . . . . . . 11  |-  ( ~P A  i^i  Fin )  C_ 
~P A
1918sseli 3344 . . . . . . . . . 10  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  ~P A )
2019elpwid 3808 . . . . . . . . 9  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  C_  A )
2120adantl 453 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  C_  A )
22 fof 5653 . . . . . . . . 9  |-  ( F : A -onto-> B  ->  F : A --> B )
23 fdm 5595 . . . . . . . . 9  |-  ( F : A --> B  ->  dom  F  =  A )
2415, 22, 233syl 19 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  dom  F  =  A )
2521, 24sseqtr4d 3385 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  C_ 
dom  F )
26 fores 5662 . . . . . . 7  |-  ( ( Fun  F  /\  x  C_ 
dom  F )  -> 
( F  |`  x
) : x -onto-> ( F " x ) )
2717, 25, 26syl2anc 643 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  ( F  |`  x ) : x -onto-> ( F "
x ) )
28 fodomnum 7938 . . . . . 6  |-  ( x  e.  dom  card  ->  ( ( F  |`  x
) : x -onto-> ( F " x )  ->  ( F "
x )  ~<_  x ) )
2914, 27, 28sylc 58 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  ( F " x )  ~<_  x )
30 simpl1 960 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  A  e.  V )
31 ssdomg 7153 . . . . . 6  |-  ( A  e.  V  ->  (
x  C_  A  ->  x  ~<_  A ) )
3230, 21, 31sylc 58 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  ~<_  A )
33 domtr 7160 . . . . 5  |-  ( ( ( F " x
)  ~<_  x  /\  x  ~<_  A )  ->  ( F " x )  ~<_  A )
3429, 32, 33syl2anc 643 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  ( F " x )  ~<_  A )
35 breq1 4215 . . . 4  |-  ( ( F " x )  =  B  ->  (
( F " x
)  ~<_  A  <->  B  ~<_  A ) )
3634, 35syl5ibcom 212 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B
)  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
( F " x
)  =  B  ->  B  ~<_  A ) )
3736rexlimdva 2830 . 2  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  ( E. x  e.  ( ~P A  i^i  Fin )
( F " x
)  =  B  ->  B  ~<_  A ) )
389, 37mpd 15 1  |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  B  ~<_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2706    i^i cin 3319    C_ wss 3320   ~Pcpw 3799   class class class wbr 4212   dom cdm 4878   ran crn 4879    |` cres 4880   "cima 4881   Fun wfun 5448    Fn wfn 5449   -->wf 5450   -onto->wfo 5452    ~<_ cdom 7107   Fincfn 7109   cardccrd 7822
This theorem is referenced by:  wdomfil  7942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-1o 6724  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-fin 7113  df-card 7826  df-acn 7829
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