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Theorem fodomb 8397
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 5646 . . . . . . . . . . . 12  |-  ( f : A -onto-> B  -> 
f : A --> B )
2 fdm 5588 . . . . . . . . . . . 12  |-  ( f : A --> B  ->  dom  f  =  A
)
31, 2syl 16 . . . . . . . . . . 11  |-  ( f : A -onto-> B  ->  dom  f  =  A
)
43eqeq1d 2444 . . . . . . . . . 10  |-  ( f : A -onto-> B  -> 
( dom  f  =  (/)  <->  A  =  (/) ) )
5 dm0rn0 5079 . . . . . . . . . . 11  |-  ( dom  f  =  (/)  <->  ran  f  =  (/) )
6 forn 5649 . . . . . . . . . . . 12  |-  ( f : A -onto-> B  ->  ran  f  =  B
)
76eqeq1d 2444 . . . . . . . . . . 11  |-  ( f : A -onto-> B  -> 
( ran  f  =  (/)  <->  B  =  (/) ) )
85, 7syl5bb 249 . . . . . . . . . 10  |-  ( f : A -onto-> B  -> 
( dom  f  =  (/)  <->  B  =  (/) ) )
94, 8bitr3d 247 . . . . . . . . 9  |-  ( f : A -onto-> B  -> 
( A  =  (/)  <->  B  =  (/) ) )
109necon3bid 2634 . . . . . . . 8  |-  ( f : A -onto-> B  -> 
( A  =/=  (/)  <->  B  =/=  (/) ) )
1110biimpac 473 . . . . . . 7  |-  ( ( A  =/=  (/)  /\  f : A -onto-> B )  ->  B  =/=  (/) )
12 vex 2952 . . . . . . . . . . . 12  |-  f  e. 
_V
1312dmex 5125 . . . . . . . . . . 11  |-  dom  f  e.  _V
143, 13syl6eqelr 2525 . . . . . . . . . 10  |-  ( f : A -onto-> B  ->  A  e.  _V )
15 fornex 5963 . . . . . . . . . 10  |-  ( A  e.  _V  ->  (
f : A -onto-> B  ->  B  e.  _V )
)
1614, 15mpcom 34 . . . . . . . . 9  |-  ( f : A -onto-> B  ->  B  e.  _V )
17 0sdomg 7229 . . . . . . . . 9  |-  ( B  e.  _V  ->  ( (/) 
~<  B  <->  B  =/=  (/) ) )
1816, 17syl 16 . . . . . . . 8  |-  ( f : A -onto-> B  -> 
( (/)  ~<  B  <->  B  =/=  (/) ) )
1918adantl 453 . . . . . . 7  |-  ( ( A  =/=  (/)  /\  f : A -onto-> B )  ->  ( (/) 
~<  B  <->  B  =/=  (/) ) )
2011, 19mpbird 224 . . . . . 6  |-  ( ( A  =/=  (/)  /\  f : A -onto-> B )  ->  (/)  ~<  B )
2120ex 424 . . . . 5  |-  ( A  =/=  (/)  ->  ( f : A -onto-> B  ->  (/)  ~<  B ) )
22 fodomg 8396 . . . . . . 7  |-  ( A  e.  _V  ->  (
f : A -onto-> B  ->  B  ~<_  A ) )
2314, 22mpcom 34 . . . . . 6  |-  ( f : A -onto-> B  ->  B  ~<_  A )
2423a1i 11 . . . . 5  |-  ( A  =/=  (/)  ->  ( f : A -onto-> B  ->  B  ~<_  A ) )
2521, 24jcad 520 . . . 4  |-  ( A  =/=  (/)  ->  ( f : A -onto-> B  ->  ( (/)  ~<  B  /\  B  ~<_  A ) ) )
2625exlimdv 1646 . . 3  |-  ( A  =/=  (/)  ->  ( E. f  f : A -onto-> B  ->  ( (/)  ~<  B  /\  B  ~<_  A ) ) )
2726imp 419 . 2  |-  ( ( A  =/=  (/)  /\  E. f  f : A -onto-> B )  ->  ( (/) 
~<  B  /\  B  ~<_  A ) )
28 sdomdomtr 7233 . . . 4  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  (/)  ~<  A )
29 reldom 7108 . . . . . . 7  |-  Rel  ~<_
3029brrelex2i 4912 . . . . . 6  |-  ( B  ~<_  A  ->  A  e.  _V )
3130adantl 453 . . . . 5  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  A  e.  _V )
32 0sdomg 7229 . . . . 5  |-  ( A  e.  _V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
3331, 32syl 16 . . . 4  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
3428, 33mpbid 202 . . 3  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  A  =/=  (/) )
35 fodomr 7251 . . 3  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  E. f 
f : A -onto-> B
)
3634, 35jca 519 . 2  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  ( A  =/=  (/)  /\  E. f 
f : A -onto-> B
) )
3727, 36impbii 181 1  |-  ( ( A  =/=  (/)  /\  E. f  f : A -onto-> B )  <->  ( (/)  ~<  B  /\  B  ~<_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2949   (/)c0 3621   class class class wbr 4205   dom cdm 4871   ran crn 4872   -->wf 5443   -onto->wfo 5445    ~<_ cdom 7100    ~< csdm 7101
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 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-ac2 8336
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 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-se 4535  df-we 4536  df-ord 4577  df-on 4578  df-suc 4580  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-isom 5456  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-er 6898  df-map 7013  df-en 7103  df-dom 7104  df-sdom 7105  df-card 7819  df-acn 7822  df-ac 7990
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