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Theorem fodomfib 7583
Description: Equivalence of an onto mapping and dominance for a nonempty finite set. Unlike fodomb 8685 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.)
Assertion
Ref Expression
fodomfib  |-  ( A  e.  Fin  ->  (
( A  =/=  (/)  /\  E. f  f : A -onto-> B )  <->  ( (/)  ~<  B  /\  B  ~<_  A ) ) )
Distinct variable groups:    A, f    B, f

Proof of Theorem fodomfib
StepHypRef Expression
1 fof 5615 . . . . . . . . . . . . 13  |-  ( f : A -onto-> B  -> 
f : A --> B )
2 fdm 5558 . . . . . . . . . . . . 13  |-  ( f : A --> B  ->  dom  f  =  A
)
31, 2syl 16 . . . . . . . . . . . 12  |-  ( f : A -onto-> B  ->  dom  f  =  A
)
43eqeq1d 2446 . . . . . . . . . . 11  |-  ( f : A -onto-> B  -> 
( dom  f  =  (/)  <->  A  =  (/) ) )
5 dm0rn0 5051 . . . . . . . . . . . 12  |-  ( dom  f  =  (/)  <->  ran  f  =  (/) )
6 forn 5618 . . . . . . . . . . . . 13  |-  ( f : A -onto-> B  ->  ran  f  =  B
)
76eqeq1d 2446 . . . . . . . . . . . 12  |-  ( f : A -onto-> B  -> 
( ran  f  =  (/)  <->  B  =  (/) ) )
85, 7syl5bb 257 . . . . . . . . . . 11  |-  ( f : A -onto-> B  -> 
( dom  f  =  (/)  <->  B  =  (/) ) )
94, 8bitr3d 255 . . . . . . . . . 10  |-  ( f : A -onto-> B  -> 
( A  =  (/)  <->  B  =  (/) ) )
109necon3bid 2638 . . . . . . . . 9  |-  ( f : A -onto-> B  -> 
( A  =/=  (/)  <->  B  =/=  (/) ) )
1110biimpac 486 . . . . . . . 8  |-  ( ( A  =/=  (/)  /\  f : A -onto-> B )  ->  B  =/=  (/) )
1211adantll 713 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/) )  /\  f : A -onto-> B )  ->  B  =/=  (/) )
13 vex 2970 . . . . . . . . . . . 12  |-  f  e. 
_V
1413rnex 6507 . . . . . . . . . . 11  |-  ran  f  e.  _V
156, 14syl6eqelr 2527 . . . . . . . . . 10  |-  ( f : A -onto-> B  ->  B  e.  _V )
1615adantl 466 . . . . . . . . 9  |-  ( ( A  e.  Fin  /\  f : A -onto-> B )  ->  B  e.  _V )
17 0sdomg 7432 . . . . . . . . 9  |-  ( B  e.  _V  ->  ( (/) 
~<  B  <->  B  =/=  (/) ) )
1816, 17syl 16 . . . . . . . 8  |-  ( ( A  e.  Fin  /\  f : A -onto-> B )  ->  ( (/)  ~<  B  <->  B  =/=  (/) ) )
1918adantlr 714 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/) )  /\  f : A -onto-> B )  ->  ( (/)  ~<  B  <->  B  =/=  (/) ) )
2012, 19mpbird 232 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/) )  /\  f : A -onto-> B )  ->  (/)  ~<  B )
2120ex 434 . . . . 5  |-  ( ( A  e.  Fin  /\  A  =/=  (/) )  ->  (
f : A -onto-> B  -> 
(/)  ~<  B ) )
22 fodomfi 7582 . . . . . . 7  |-  ( ( A  e.  Fin  /\  f : A -onto-> B )  ->  B  ~<_  A )
2322ex 434 . . . . . 6  |-  ( A  e.  Fin  ->  (
f : A -onto-> B  ->  B  ~<_  A ) )
2423adantr 465 . . . . 5  |-  ( ( A  e.  Fin  /\  A  =/=  (/) )  ->  (
f : A -onto-> B  ->  B  ~<_  A ) )
2521, 24jcad 533 . . . 4  |-  ( ( A  e.  Fin  /\  A  =/=  (/) )  ->  (
f : A -onto-> B  ->  ( (/)  ~<  B  /\  B  ~<_  A ) ) )
2625exlimdv 1690 . . 3  |-  ( ( A  e.  Fin  /\  A  =/=  (/) )  ->  ( E. f  f : A -onto-> B  ->  ( (/)  ~<  B  /\  B  ~<_  A ) ) )
2726expimpd 603 . 2  |-  ( A  e.  Fin  ->  (
( A  =/=  (/)  /\  E. f  f : A -onto-> B )  ->  ( (/) 
~<  B  /\  B  ~<_  A ) ) )
28 sdomdomtr 7436 . . . 4  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  (/)  ~<  A )
29 0sdomg 7432 . . . 4  |-  ( A  e.  Fin  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
3028, 29syl5ib 219 . . 3  |-  ( A  e.  Fin  ->  (
( (/)  ~<  B  /\  B  ~<_  A )  ->  A  =/=  (/) ) )
31 fodomr 7454 . . . 4  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  E. f 
f : A -onto-> B
)
3231a1i 11 . . 3  |-  ( A  e.  Fin  ->  (
( (/)  ~<  B  /\  B  ~<_  A )  ->  E. f  f : A -onto-> B ) )
3330, 32jcad 533 . 2  |-  ( A  e.  Fin  ->  (
( (/)  ~<  B  /\  B  ~<_  A )  -> 
( A  =/=  (/)  /\  E. f  f : A -onto-> B ) ) )
3427, 33impbid 191 1  |-  ( A  e.  Fin  ->  (
( A  =/=  (/)  /\  E. f  f : A -onto-> B )  <->  ( (/)  ~<  B  /\  B  ~<_  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2601   _Vcvv 2967   (/)c0 3632   class class class wbr 4287   dom cdm 4835   ran crn 4836   -->wf 5409   -onto->wfo 5411    ~<_ cdom 7300    ~< csdm 7301   Fincfn 7302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-om 6472  df-1o 6912  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306
This theorem is referenced by: (None)
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