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Theorem fodomb 8954
Description: Equivalence of an onto mapping and dominance for a nonempty 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 5793 . . . . . . . . . . . 12  |-  ( f : A -onto-> B  -> 
f : A --> B )
2 fdm 5733 . . . . . . . . . . . 12  |-  ( f : A --> B  ->  dom  f  =  A
)
31, 2syl 17 . . . . . . . . . . 11  |-  ( f : A -onto-> B  ->  dom  f  =  A
)
43eqeq1d 2453 . . . . . . . . . 10  |-  ( f : A -onto-> B  -> 
( dom  f  =  (/)  <->  A  =  (/) ) )
5 dm0rn0 5051 . . . . . . . . . . 11  |-  ( dom  f  =  (/)  <->  ran  f  =  (/) )
6 forn 5796 . . . . . . . . . . . 12  |-  ( f : A -onto-> B  ->  ran  f  =  B
)
76eqeq1d 2453 . . . . . . . . . . 11  |-  ( f : A -onto-> B  -> 
( ran  f  =  (/)  <->  B  =  (/) ) )
85, 7syl5bb 261 . . . . . . . . . 10  |-  ( f : A -onto-> B  -> 
( dom  f  =  (/)  <->  B  =  (/) ) )
94, 8bitr3d 259 . . . . . . . . 9  |-  ( f : A -onto-> B  -> 
( A  =  (/)  <->  B  =  (/) ) )
109necon3bid 2668 . . . . . . . 8  |-  ( f : A -onto-> B  -> 
( A  =/=  (/)  <->  B  =/=  (/) ) )
1110biimpac 489 . . . . . . 7  |-  ( ( A  =/=  (/)  /\  f : A -onto-> B )  ->  B  =/=  (/) )
12 vex 3048 . . . . . . . . . . . 12  |-  f  e. 
_V
1312dmex 6726 . . . . . . . . . . 11  |-  dom  f  e.  _V
143, 13syl6eqelr 2538 . . . . . . . . . 10  |-  ( f : A -onto-> B  ->  A  e.  _V )
15 fornex 6762 . . . . . . . . . 10  |-  ( A  e.  _V  ->  (
f : A -onto-> B  ->  B  e.  _V )
)
1614, 15mpcom 37 . . . . . . . . 9  |-  ( f : A -onto-> B  ->  B  e.  _V )
17 0sdomg 7701 . . . . . . . . 9  |-  ( B  e.  _V  ->  ( (/) 
~<  B  <->  B  =/=  (/) ) )
1816, 17syl 17 . . . . . . . 8  |-  ( f : A -onto-> B  -> 
( (/)  ~<  B  <->  B  =/=  (/) ) )
1918adantl 468 . . . . . . 7  |-  ( ( A  =/=  (/)  /\  f : A -onto-> B )  ->  ( (/) 
~<  B  <->  B  =/=  (/) ) )
2011, 19mpbird 236 . . . . . 6  |-  ( ( A  =/=  (/)  /\  f : A -onto-> B )  ->  (/)  ~<  B )
2120ex 436 . . . . 5  |-  ( A  =/=  (/)  ->  ( f : A -onto-> B  ->  (/)  ~<  B ) )
22 fodomg 8953 . . . . . . 7  |-  ( A  e.  _V  ->  (
f : A -onto-> B  ->  B  ~<_  A ) )
2314, 22mpcom 37 . . . . . 6  |-  ( f : A -onto-> B  ->  B  ~<_  A )
2423a1i 11 . . . . 5  |-  ( A  =/=  (/)  ->  ( f : A -onto-> B  ->  B  ~<_  A ) )
2521, 24jcad 536 . . . 4  |-  ( A  =/=  (/)  ->  ( f : A -onto-> B  ->  ( (/)  ~<  B  /\  B  ~<_  A ) ) )
2625exlimdv 1779 . . 3  |-  ( A  =/=  (/)  ->  ( E. f  f : A -onto-> B  ->  ( (/)  ~<  B  /\  B  ~<_  A ) ) )
2726imp 431 . 2  |-  ( ( A  =/=  (/)  /\  E. f  f : A -onto-> B )  ->  ( (/) 
~<  B  /\  B  ~<_  A ) )
28 sdomdomtr 7705 . . . 4  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  (/)  ~<  A )
29 reldom 7575 . . . . . . 7  |-  Rel  ~<_
3029brrelex2i 4876 . . . . . 6  |-  ( B  ~<_  A  ->  A  e.  _V )
3130adantl 468 . . . . 5  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  A  e.  _V )
32 0sdomg 7701 . . . . 5  |-  ( A  e.  _V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
3331, 32syl 17 . . . 4  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
3428, 33mpbid 214 . . 3  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  A  =/=  (/) )
35 fodomr 7723 . . 3  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  E. f 
f : A -onto-> B
)
3634, 35jca 535 . 2  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  ( A  =/=  (/)  /\  E. f 
f : A -onto-> B
) )
3727, 36impbii 191 1  |-  ( ( A  =/=  (/)  /\  E. f  f : A -onto-> B )  <->  ( (/)  ~<  B  /\  B  ~<_  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444   E.wex 1663    e. wcel 1887    =/= wne 2622   _Vcvv 3045   (/)c0 3731   class class class wbr 4402   dom cdm 4834   ran crn 4835   -->wf 5578   -onto->wfo 5580    ~<_ cdom 7567    ~< csdm 7568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-ac2 8893
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-card 8373  df-acn 8376  df-ac 8547
This theorem is referenced by: (None)
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