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Theorem fodomfib 7818
Description: Equivalence of an onto mapping and dominance for a nonempty finite set. Unlike fodomb 8921 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 5801 . . . . . . . . . . . . 13  |-  ( f : A -onto-> B  -> 
f : A --> B )
2 fdm 5741 . . . . . . . . . . . . 13  |-  ( f : A --> B  ->  dom  f  =  A
)
31, 2syl 16 . . . . . . . . . . . 12  |-  ( f : A -onto-> B  ->  dom  f  =  A
)
43eqeq1d 2459 . . . . . . . . . . 11  |-  ( f : A -onto-> B  -> 
( dom  f  =  (/)  <->  A  =  (/) ) )
5 dm0rn0 5229 . . . . . . . . . . . 12  |-  ( dom  f  =  (/)  <->  ran  f  =  (/) )
6 forn 5804 . . . . . . . . . . . . 13  |-  ( f : A -onto-> B  ->  ran  f  =  B
)
76eqeq1d 2459 . . . . . . . . . . . 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 2715 . . . . . . . . 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 3112 . . . . . . . . . . . 12  |-  f  e. 
_V
1413rnex 6733 . . . . . . . . . . 11  |-  ran  f  e.  _V
156, 14syl6eqelr 2554 . . . . . . . . . 10  |-  ( f : A -onto-> B  ->  B  e.  _V )
1615adantl 466 . . . . . . . . 9  |-  ( ( A  e.  Fin  /\  f : A -onto-> B )  ->  B  e.  _V )
17 0sdomg 7665 . . . . . . . . 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 7817 . . . . . . 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 1725 . . 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 7669 . . . 4  |-  ( (
(/)  ~<  B  /\  B  ~<_  A )  ->  (/)  ~<  A )
29 0sdomg 7665 . . . 4  |-  ( A  e.  Fin  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
3028, 29syl5ib 219 . . 3  |-  ( A  e.  Fin  ->  (
( (/)  ~<  B  /\  B  ~<_  A )  ->  A  =/=  (/) ) )
31 fodomr 7687 . . . 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 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   _Vcvv 3109   (/)c0 3793   class class class wbr 4456   dom cdm 5008   ran crn 5009   -->wf 5590   -onto->wfo 5592    ~<_ cdom 7533    ~< csdm 7534   Fincfn 7535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-1o 7148  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539
This theorem is referenced by: (None)
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