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Theorem fodomfib 7695
Description: Equivalence of an onto mapping and dominance for a nonempty finite set. Unlike fodomb 8797 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 5721 . . . . . . . . . . . . 13  |-  ( f : A -onto-> B  -> 
f : A --> B )
2 fdm 5664 . . . . . . . . . . . . 13  |-  ( f : A --> B  ->  dom  f  =  A
)
31, 2syl 16 . . . . . . . . . . . 12  |-  ( f : A -onto-> B  ->  dom  f  =  A
)
43eqeq1d 2453 . . . . . . . . . . 11  |-  ( f : A -onto-> B  -> 
( dom  f  =  (/)  <->  A  =  (/) ) )
5 dm0rn0 5157 . . . . . . . . . . . 12  |-  ( dom  f  =  (/)  <->  ran  f  =  (/) )
6 forn 5724 . . . . . . . . . . . . 13  |-  ( f : A -onto-> B  ->  ran  f  =  B
)
76eqeq1d 2453 . . . . . . . . . . . 12  |-  ( f : A -onto-> B  -> 
( ran  f  =  (/)  <->  B  =  (/) ) )
85, 7syl5bb 257 . . . . . . . . . . 11  |-  ( f : A -onto-> B  -> 
( dom  f  =  (/)  <->  B  =  (/) ) )
94, 8bitr3d 255 . . . . . . . . . 10  |-  ( f : A -onto-> B  -> 
( A  =  (/)  <->  B  =  (/) ) )
109necon3bid 2706 . . . . . . . . 9  |-  ( f : A -onto-> B  -> 
( A  =/=  (/)  <->  B  =/=  (/) ) )
1110biimpac 486 . . . . . . . 8  |-  ( ( A  =/=  (/)  /\  f : A -onto-> B )  ->  B  =/=  (/) )
1211adantll 713 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/) )  /\  f : A -onto-> B )  ->  B  =/=  (/) )
13 vex 3074 . . . . . . . . . . . 12  |-  f  e. 
_V
1413rnex 6615 . . . . . . . . . . 11  |-  ran  f  e.  _V
156, 14syl6eqelr 2548 . . . . . . . . . 10  |-  ( f : A -onto-> B  ->  B  e.  _V )
1615adantl 466 . . . . . . . . 9  |-  ( ( A  e.  Fin  /\  f : A -onto-> B )  ->  B  e.  _V )
17 0sdomg 7543 . . . . . . . . 9  |-  ( B  e.  _V  ->  ( (/) 
~<  B  <->  B  =/=  (/) ) )
1816, 17syl 16 . . . . . . . 8  |-  ( ( A  e.  Fin  /\  f : A -onto-> B )  ->  ( (/)  ~<  B  <->  B  =/=  (/) ) )
1918adantlr 714 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/) )  /\  f : A -onto-> B )  ->  ( (/)  ~<  B  <->  B  =/=  (/) ) )
2012, 19mpbird 232 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  A  =/=  (/) )  /\  f : A -onto-> B )  ->  (/)  ~<  B )
2120ex 434 . . . . 5  |-  ( ( A  e.  Fin  /\  A  =/=  (/) )  ->  (
f : A -onto-> B  -> 
(/)  ~<  B ) )
22 fodomfi 7694 . . . . . . 7  |-  ( ( A  e.  Fin  /\  f : A -onto-> B )  ->  B  ~<_  A )
2322ex 434 . . . . . 6  |-  ( A  e.  Fin  ->  (
f : A -onto-> B  ->  B  ~<_  A ) )
2423adantr 465 . . . . 5  |-  ( ( A  e.  Fin  /\  A  =/=  (/) )  ->  (
f : A -onto-> B  ->  B  ~<_  A ) )
2521, 24jcad 533 . . . 4  |-  ( ( A  e.  Fin  /\  A  =/=  (/) )  ->  (
f : A -onto-> B  ->  ( (/)  ~<  B  /\  B  ~<_  A ) ) )
2625exlimdv 1691 . . 3  |-  ( ( A  e.  Fin  /\  A  =/=  (/) )  ->  ( E. f  f : A -onto-> B  ->  ( (/)  ~<  B  /\  B  ~<_  A ) ) )
2726expimpd 603 . 2  |-  ( A  e.  Fin  ->  (
( A  =/=  (/)  /\  E. f  f : A -onto-> B )  ->  ( (/) 
~<  B  /\  B  ~<_  A ) ) )
28 sdomdomtr 7547 . . . 4  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  (/)  ~<  A )
29 0sdomg 7543 . . . 4  |-  ( A  e.  Fin  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
3028, 29syl5ib 219 . . 3  |-  ( A  e.  Fin  ->  (
( (/)  ~<  B  /\  B  ~<_  A )  ->  A  =/=  (/) ) )
31 fodomr 7565 . . . 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 533 . 2  |-  ( A  e.  Fin  ->  (
( (/)  ~<  B  /\  B  ~<_  A )  -> 
( A  =/=  (/)  /\  E. f  f : A -onto-> B ) ) )
3427, 33impbid 191 1  |-  ( A  e.  Fin  ->  (
( A  =/=  (/)  /\  E. f  f : A -onto-> B )  <->  ( (/)  ~<  B  /\  B  ~<_  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2644   _Vcvv 3071   (/)c0 3738   class class class wbr 4393   dom cdm 4941   ran crn 4942   -->wf 5515   -onto->wfo 5517    ~<_ cdom 7411    ~< csdm 7412   Fincfn 7413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-om 6580  df-1o 7023  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417
This theorem is referenced by: (None)
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