MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fodomfib Structured version   Unicode version

Theorem fodomfib 7387
Description: Equivalence of an onto mapping and dominance for a non-empty finite set. Unlike fodomb 8405 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 5654 . . . . . . . . . . . . 13  |-  ( f : A -onto-> B  -> 
f : A --> B )
2 fdm 5596 . . . . . . . . . . . . 13  |-  ( f : A --> B  ->  dom  f  =  A
)
31, 2syl 16 . . . . . . . . . . . 12  |-  ( f : A -onto-> B  ->  dom  f  =  A
)
43eqeq1d 2445 . . . . . . . . . . 11  |-  ( f : A -onto-> B  -> 
( dom  f  =  (/)  <->  A  =  (/) ) )
5 dm0rn0 5087 . . . . . . . . . . . 12  |-  ( dom  f  =  (/)  <->  ran  f  =  (/) )
6 forn 5657 . . . . . . . . . . . . 13  |-  ( f : A -onto-> B  ->  ran  f  =  B
)
76eqeq1d 2445 . . . . . . . . . . . 12  |-  ( f : A -onto-> B  -> 
( ran  f  =  (/)  <->  B  =  (/) ) )
85, 7syl5bb 250 . . . . . . . . . . 11  |-  ( f : A -onto-> B  -> 
( dom  f  =  (/)  <->  B  =  (/) ) )
94, 8bitr3d 248 . . . . . . . . . 10  |-  ( f : A -onto-> B  -> 
( A  =  (/)  <->  B  =  (/) ) )
109necon3bid 2637 . . . . . . . . 9  |-  ( f : A -onto-> B  -> 
( A  =/=  (/)  <->  B  =/=  (/) ) )
1110biimpac 474 . . . . . . . 8  |-  ( ( A  =/=  (/)  /\  f : A -onto-> B )  ->  B  =/=  (/) )
1211adantll 696 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/) )  /\  f : A -onto-> B )  ->  B  =/=  (/) )
13 vex 2960 . . . . . . . . . . . 12  |-  f  e. 
_V
1413rnex 5134 . . . . . . . . . . 11  |-  ran  f  e.  _V
156, 14syl6eqelr 2526 . . . . . . . . . 10  |-  ( f : A -onto-> B  ->  B  e.  _V )
1615adantl 454 . . . . . . . . 9  |-  ( ( A  e.  Fin  /\  f : A -onto-> B )  ->  B  e.  _V )
17 0sdomg 7237 . . . . . . . . 9  |-  ( B  e.  _V  ->  ( (/) 
~<  B  <->  B  =/=  (/) ) )
1816, 17syl 16 . . . . . . . 8  |-  ( ( A  e.  Fin  /\  f : A -onto-> B )  ->  ( (/)  ~<  B  <->  B  =/=  (/) ) )
1918adantlr 697 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/) )  /\  f : A -onto-> B )  ->  ( (/)  ~<  B  <->  B  =/=  (/) ) )
2012, 19mpbird 225 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/) )  /\  f : A -onto-> B )  ->  (/)  ~<  B )
2120ex 425 . . . . 5  |-  ( ( A  e.  Fin  /\  A  =/=  (/) )  ->  (
f : A -onto-> B  -> 
(/)  ~<  B ) )
22 fodomfi 7386 . . . . . . 7  |-  ( ( A  e.  Fin  /\  f : A -onto-> B )  ->  B  ~<_  A )
2322ex 425 . . . . . 6  |-  ( A  e.  Fin  ->  (
f : A -onto-> B  ->  B  ~<_  A ) )
2423adantr 453 . . . . 5  |-  ( ( A  e.  Fin  /\  A  =/=  (/) )  ->  (
f : A -onto-> B  ->  B  ~<_  A ) )
2521, 24jcad 521 . . . 4  |-  ( ( A  e.  Fin  /\  A  =/=  (/) )  ->  (
f : A -onto-> B  ->  ( (/)  ~<  B  /\  B  ~<_  A ) ) )
2625exlimdv 1647 . . 3  |-  ( ( A  e.  Fin  /\  A  =/=  (/) )  ->  ( E. f  f : A -onto-> B  ->  ( (/)  ~<  B  /\  B  ~<_  A ) ) )
2726expimpd 588 . 2  |-  ( A  e.  Fin  ->  (
( A  =/=  (/)  /\  E. f  f : A -onto-> B )  ->  ( (/) 
~<  B  /\  B  ~<_  A ) ) )
28 sdomdomtr 7241 . . . 4  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  (/)  ~<  A )
29 0sdomg 7237 . . . 4  |-  ( A  e.  Fin  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
3028, 29syl5ib 212 . . 3  |-  ( A  e.  Fin  ->  (
( (/)  ~<  B  /\  B  ~<_  A )  ->  A  =/=  (/) ) )
31 fodomr 7259 . . . 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 521 . 2  |-  ( A  e.  Fin  ->  (
( (/)  ~<  B  /\  B  ~<_  A )  -> 
( A  =/=  (/)  /\  E. f  f : A -onto-> B ) ) )
3427, 33impbid 185 1  |-  ( A  e.  Fin  ->  (
( A  =/=  (/)  /\  E. f  f : A -onto-> B )  <->  ( (/)  ~<  B  /\  B  ~<_  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2600   _Vcvv 2957   (/)c0 3629   class class class wbr 4213   dom cdm 4879   ran crn 4880   -->wf 5451   -onto->wfo 5453    ~<_ cdom 7108    ~< csdm 7109   Fincfn 7110
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-1o 6725  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114
  Copyright terms: Public domain W3C validator