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

Theorem fodomb 8147
Description: Equivalence of an onto mapping and dominance for a non-empty set. Proposition 10.35 of [TakeutiZaring] p. 93. (Contributed by NM, 29-Jul-2004.)
Assertion
Ref Expression
fodomb  |-  ( ( A  =/=  (/)  /\  E. f  f : A -onto-> B )  <->  ( (/)  ~<  B  /\  B  ~<_  A ) )
Distinct variable groups:    A, f    B, f

Proof of Theorem fodomb
StepHypRef Expression
1 fof 5417 . . . . . . . . . . . 12  |-  ( f : A -onto-> B  -> 
f : A --> B )
2 fdm 5359 . . . . . . . . . . . 12  |-  ( f : A --> B  ->  dom  f  =  A
)
31, 2syl 15 . . . . . . . . . . 11  |-  ( f : A -onto-> B  ->  dom  f  =  A
)
43eqeq1d 2292 . . . . . . . . . 10  |-  ( f : A -onto-> B  -> 
(  dom  f  =  (/)  <->  A  =  (/) ) )
5 dm0rn0 4894 . . . . . . . . . . 11  |-  (  dom  f  =  (/)  <->  ran  f  =  (/) )
6 forn 5420 . . . . . . . . . . . 12  |-  ( f : A -onto-> B  ->  ran  f  =  B
)
76eqeq1d 2292 . . . . . . . . . . 11  |-  ( f : A -onto-> B  -> 
( ran  f  =  (/)  <->  B  =  (/) ) )
85, 7syl5bb 248 . . . . . . . . . 10  |-  ( f : A -onto-> B  -> 
(  dom  f  =  (/)  <->  B  =  (/) ) )
94, 8bitr3d 246 . . . . . . . . 9  |-  ( f : A -onto-> B  -> 
( A  =  (/)  <->  B  =  (/) ) )
109necon3bid 2482 . . . . . . . 8  |-  ( f : A -onto-> B  -> 
( A  =/=  (/)  <->  B  =/=  (/) ) )
1110biimpac 472 . . . . . . 7  |-  ( ( A  =/=  (/)  /\  f : A -onto-> B )  ->  B  =/=  (/) )
12 vex 2792 . . . . . . . . . . . 12  |-  f  e. 
_V
1312dmex 4940 . . . . . . . . . . 11  |-  dom  f  e.  _V
143, 13syl6eqelr 2373 . . . . . . . . . 10  |-  ( f : A -onto-> B  ->  A  e.  _V )
15 fornex 5712 . . . . . . . . . 10  |-  ( A  e.  _V  ->  (
f : A -onto-> B  ->  B  e.  _V )
)
1614, 15mpcom 32 . . . . . . . . 9  |-  ( f : A -onto-> B  ->  B  e.  _V )
17 0sdomg 6986 . . . . . . . . 9  |-  ( B  e.  _V  ->  ( (/) 
~<  B  <->  B  =/=  (/) ) )
1816, 17syl 15 . . . . . . . 8  |-  ( f : A -onto-> B  -> 
( (/)  ~<  B  <->  B  =/=  (/) ) )
1918adantl 452 . . . . . . 7  |-  ( ( A  =/=  (/)  /\  f : A -onto-> B )  ->  ( (/) 
~<  B  <->  B  =/=  (/) ) )
2011, 19mpbird 223 . . . . . 6  |-  ( ( A  =/=  (/)  /\  f : A -onto-> B )  ->  (/)  ~<  B )
2120ex 423 . . . . 5  |-  ( A  =/=  (/)  ->  ( f : A -onto-> B  ->  (/)  ~<  B ) )
22 fodomg 8146 . . . . . . 7  |-  ( A  e.  _V  ->  (
f : A -onto-> B  ->  B  ~<_  A ) )
2314, 22mpcom 32 . . . . . 6  |-  ( f : A -onto-> B  ->  B  ~<_  A )
2423a1i 10 . . . . 5  |-  ( A  =/=  (/)  ->  ( f : A -onto-> B  ->  B  ~<_  A ) )
2521, 24jcad 519 . . . 4  |-  ( A  =/=  (/)  ->  ( f : A -onto-> B  ->  ( (/)  ~<  B  /\  B  ~<_  A ) ) )
2625exlimdv 1665 . . 3  |-  ( A  =/=  (/)  ->  ( E. f  f : A -onto-> B  ->  ( (/)  ~<  B  /\  B  ~<_  A ) ) )
2726imp 418 . 2  |-  ( ( A  =/=  (/)  /\  E. f  f : A -onto-> B )  ->  ( (/) 
~<  B  /\  B  ~<_  A ) )
28 sdomdomtr 6990 . . . 4  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  (/)  ~<  A )
29 reldom 6865 . . . . . . 7  |-  Rel  ~<_
3029brrelex2i 4729 . . . . . 6  |-  ( B  ~<_  A  ->  A  e.  _V )
3130adantl 452 . . . . 5  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  A  e.  _V )
32 0sdomg 6986 . . . . 5  |-  ( A  e.  _V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
3331, 32syl 15 . . . 4  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
3428, 33mpbid 201 . . 3  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  A  =/=  (/) )
35 fodomr 7008 . . 3  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  E. f 
f : A -onto-> B
)
3634, 35jca 518 . 2  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  ( A  =/=  (/)  /\  E. f 
f : A -onto-> B
) )
3727, 36impbii 180 1  |-  ( ( 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 1685    =/= wne 2447   _Vcvv 2789   (/)c0 3456   class class class wbr 4024    dom cdm 4688   ran crn 4689   -->wf 5217   -onto->wfo 5219    ~<_ cdom 6857    ~< csdm 6858
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-ac2 8085
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-card 7568  df-acn 7571  df-ac 7739
  Copyright terms: Public domain W3C validator