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

Theorem fodomfib 7136
Description: Equivalence of an onto mapping and dominance for a non-empty finite set. Unlike fodomb 8151 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 5451 . . . . . . . . . . . . 13  |-  ( f : A -onto-> B  -> 
f : A --> B )
2 fdm 5393 . . . . . . . . . . . . 13  |-  ( f : A --> B  ->  dom  f  =  A
)
31, 2syl 15 . . . . . . . . . . . 12  |-  ( f : A -onto-> B  ->  dom  f  =  A
)
43eqeq1d 2291 . . . . . . . . . . 11  |-  ( f : A -onto-> B  -> 
( dom  f  =  (/)  <->  A  =  (/) ) )
5 dm0rn0 4895 . . . . . . . . . . . 12  |-  ( dom  f  =  (/)  <->  ran  f  =  (/) )
6 forn 5454 . . . . . . . . . . . . 13  |-  ( f : A -onto-> B  ->  ran  f  =  B
)
76eqeq1d 2291 . . . . . . . . . . . 12  |-  ( f : A -onto-> B  -> 
( ran  f  =  (/)  <->  B  =  (/) ) )
85, 7syl5bb 248 . . . . . . . . . . 11  |-  ( f : A -onto-> B  -> 
( dom  f  =  (/)  <->  B  =  (/) ) )
94, 8bitr3d 246 . . . . . . . . . 10  |-  ( f : A -onto-> B  -> 
( A  =  (/)  <->  B  =  (/) ) )
109necon3bid 2481 . . . . . . . . 9  |-  ( f : A -onto-> B  -> 
( A  =/=  (/)  <->  B  =/=  (/) ) )
1110biimpac 472 . . . . . . . 8  |-  ( ( A  =/=  (/)  /\  f : A -onto-> B )  ->  B  =/=  (/) )
1211adantll 694 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/) )  /\  f : A -onto-> B )  ->  B  =/=  (/) )
13 vex 2791 . . . . . . . . . . . 12  |-  f  e. 
_V
1413rnex 4942 . . . . . . . . . . 11  |-  ran  f  e.  _V
156, 14syl6eqelr 2372 . . . . . . . . . 10  |-  ( f : A -onto-> B  ->  B  e.  _V )
1615adantl 452 . . . . . . . . 9  |-  ( ( A  e.  Fin  /\  f : A -onto-> B )  ->  B  e.  _V )
17 0sdomg 6990 . . . . . . . . 9  |-  ( B  e.  _V  ->  ( (/) 
~<  B  <->  B  =/=  (/) ) )
1816, 17syl 15 . . . . . . . 8  |-  ( ( A  e.  Fin  /\  f : A -onto-> B )  ->  ( (/)  ~<  B  <->  B  =/=  (/) ) )
1918adantlr 695 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/) )  /\  f : A -onto-> B )  ->  ( (/)  ~<  B  <->  B  =/=  (/) ) )
2012, 19mpbird 223 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/) )  /\  f : A -onto-> B )  ->  (/)  ~<  B )
2120ex 423 . . . . 5  |-  ( ( A  e.  Fin  /\  A  =/=  (/) )  ->  (
f : A -onto-> B  -> 
(/)  ~<  B ) )
22 fodomfi 7135 . . . . . . 7  |-  ( ( A  e.  Fin  /\  f : A -onto-> B )  ->  B  ~<_  A )
2322ex 423 . . . . . 6  |-  ( A  e.  Fin  ->  (
f : A -onto-> B  ->  B  ~<_  A ) )
2423adantr 451 . . . . 5  |-  ( ( A  e.  Fin  /\  A  =/=  (/) )  ->  (
f : A -onto-> B  ->  B  ~<_  A ) )
2521, 24jcad 519 . . . 4  |-  ( ( A  e.  Fin  /\  A  =/=  (/) )  ->  (
f : A -onto-> B  ->  ( (/)  ~<  B  /\  B  ~<_  A ) ) )
2625exlimdv 1664 . . 3  |-  ( ( A  e.  Fin  /\  A  =/=  (/) )  ->  ( E. f  f : A -onto-> B  ->  ( (/)  ~<  B  /\  B  ~<_  A ) ) )
2726expimpd 586 . 2  |-  ( A  e.  Fin  ->  (
( A  =/=  (/)  /\  E. f  f : A -onto-> B )  ->  ( (/) 
~<  B  /\  B  ~<_  A ) ) )
28 sdomdomtr 6994 . . . 4  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  (/)  ~<  A )
29 0sdomg 6990 . . . 4  |-  ( A  e.  Fin  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
3028, 29syl5ib 210 . . 3  |-  ( A  e.  Fin  ->  (
( (/)  ~<  B  /\  B  ~<_  A )  ->  A  =/=  (/) ) )
31 fodomr 7012 . . . 4  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  E. f 
f : A -onto-> B
)
3231a1i 10 . . 3  |-  ( A  e.  Fin  ->  (
( (/)  ~<  B  /\  B  ~<_  A )  ->  E. f  f : A -onto-> B ) )
3330, 32jcad 519 . 2  |-  ( A  e.  Fin  ->  (
( (/)  ~<  B  /\  B  ~<_  A )  -> 
( A  =/=  (/)  /\  E. f  f : A -onto-> B ) ) )
3427, 33impbid 183 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 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   (/)c0 3455   class class class wbr 4023   dom cdm 4689   ran crn 4690   -->wf 5251   -onto->wfo 5253    ~<_ cdom 6861    ~< csdm 6862   Fincfn 6863
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-reu 2550  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-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-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867
  Copyright terms: Public domain W3C validator