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Theorem fodomfi 7314
Description: An onto function implies dominance of domain over range, for finite sets. Unlike fodom 8328 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
fodomfi  |-  ( ( A  e.  Fin  /\  F : A -onto-> B )  ->  B  ~<_  A )

Proof of Theorem fodomfi
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 foima 5591 . . 3  |-  ( F : A -onto-> B  -> 
( F " A
)  =  B )
21adantl 453 . 2  |-  ( ( A  e.  Fin  /\  F : A -onto-> B )  ->  ( F " A )  =  B )
3 fofn 5588 . . . 4  |-  ( F : A -onto-> B  ->  F  Fn  A )
4 imaeq2 5132 . . . . . . . 8  |-  ( x  =  (/)  ->  ( F
" x )  =  ( F " (/) ) )
5 ima0 5154 . . . . . . . 8  |-  ( F
" (/) )  =  (/)
64, 5syl6eq 2428 . . . . . . 7  |-  ( x  =  (/)  ->  ( F
" x )  =  (/) )
7 id 20 . . . . . . 7  |-  ( x  =  (/)  ->  x  =  (/) )
86, 7breq12d 4159 . . . . . 6  |-  ( x  =  (/)  ->  ( ( F " x )  ~<_  x  <->  (/)  ~<_  (/) ) )
98imbi2d 308 . . . . 5  |-  ( x  =  (/)  ->  ( ( F  Fn  A  -> 
( F " x
)  ~<_  x )  <->  ( F  Fn  A  ->  (/)  ~<_  (/) ) ) )
10 imaeq2 5132 . . . . . . 7  |-  ( x  =  y  ->  ( F " x )  =  ( F " y
) )
11 id 20 . . . . . . 7  |-  ( x  =  y  ->  x  =  y )
1210, 11breq12d 4159 . . . . . 6  |-  ( x  =  y  ->  (
( F " x
)  ~<_  x  <->  ( F " y )  ~<_  y ) )
1312imbi2d 308 . . . . 5  |-  ( x  =  y  ->  (
( F  Fn  A  ->  ( F " x
)  ~<_  x )  <->  ( F  Fn  A  ->  ( F
" y )  ~<_  y ) ) )
14 imaeq2 5132 . . . . . . 7  |-  ( x  =  ( y  u. 
{ z } )  ->  ( F "
x )  =  ( F " ( y  u.  { z } ) ) )
15 id 20 . . . . . . 7  |-  ( x  =  ( y  u. 
{ z } )  ->  x  =  ( y  u.  { z } ) )
1614, 15breq12d 4159 . . . . . 6  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( F
" x )  ~<_  x  <-> 
( F " (
y  u.  { z } ) )  ~<_  ( y  u.  { z } ) ) )
1716imbi2d 308 . . . . 5  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( F  Fn  A  ->  ( F " x )  ~<_  x )  <->  ( F  Fn  A  ->  ( F "
( y  u.  {
z } ) )  ~<_  ( y  u.  {
z } ) ) ) )
18 imaeq2 5132 . . . . . . 7  |-  ( x  =  A  ->  ( F " x )  =  ( F " A
) )
19 id 20 . . . . . . 7  |-  ( x  =  A  ->  x  =  A )
2018, 19breq12d 4159 . . . . . 6  |-  ( x  =  A  ->  (
( F " x
)  ~<_  x  <->  ( F " A )  ~<_  A ) )
2120imbi2d 308 . . . . 5  |-  ( x  =  A  ->  (
( F  Fn  A  ->  ( F " x
)  ~<_  x )  <->  ( F  Fn  A  ->  ( F
" A )  ~<_  A ) ) )
22 0ex 4273 . . . . . . 7  |-  (/)  e.  _V
23220dom 7166 . . . . . 6  |-  (/)  ~<_  (/)
2423a1i 11 . . . . 5  |-  ( F  Fn  A  ->  (/)  ~<_  (/) )
25 fnfun 5475 . . . . . . . . . . . . . . 15  |-  ( F  Fn  A  ->  Fun  F )
2625ad2antrl 709 . . . . . . . . . . . . . 14  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  Fun  F )
27 funressn 5851 . . . . . . . . . . . . . 14  |-  ( Fun 
F  ->  ( F  |` 
{ z } ) 
C_  { <. z ,  ( F `  z ) >. } )
28 rnss 5031 . . . . . . . . . . . . . 14  |-  ( ( F  |`  { z } )  C_  { <. z ,  ( F `  z ) >. }  ->  ran  ( F  |`  { z } )  C_  ran  {
<. z ,  ( F `
 z ) >. } )
2926, 27, 283syl 19 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  ran  ( F  |`  { z } )  C_  ran  {
<. z ,  ( F `
 z ) >. } )
30 df-ima 4824 . . . . . . . . . . . . 13  |-  ( F
" { z } )  =  ran  ( F  |`  { z } )
31 vex 2895 . . . . . . . . . . . . . . 15  |-  z  e. 
_V
3231rnsnop 5283 . . . . . . . . . . . . . 14  |-  ran  { <. z ,  ( F `
 z ) >. }  =  { ( F `  z ) }
3332eqcomi 2384 . . . . . . . . . . . . 13  |-  { ( F `  z ) }  =  ran  { <. z ,  ( F `
 z ) >. }
3429, 30, 333sstr4g 3325 . . . . . . . . . . . 12  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  ( F " { z } )  C_  { ( F `  z ) } )
35 snex 4339 . . . . . . . . . . . 12  |-  { ( F `  z ) }  e.  _V
36 ssexg 4283 . . . . . . . . . . . 12  |-  ( ( ( F " {
z } )  C_  { ( F `  z
) }  /\  {
( F `  z
) }  e.  _V )  ->  ( F " { z } )  e.  _V )
3734, 35, 36sylancl 644 . . . . . . . . . . 11  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  ( F " { z } )  e.  _V )
38 fvi 5715 . . . . . . . . . . 11  |-  ( ( F " { z } )  e.  _V  ->  (  _I  `  ( F " { z } ) )  =  ( F " { z } ) )
3937, 38syl 16 . . . . . . . . . 10  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  (  _I  `  ( F " { z } ) )  =  ( F
" { z } ) )
4039uneq2d 3437 . . . . . . . . 9  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  (
( F " y
)  u.  (  _I 
`  ( F " { z } ) ) )  =  ( ( F " y
)  u.  ( F
" { z } ) ) )
41 imaundi 5217 . . . . . . . . 9  |-  ( F
" ( y  u. 
{ z } ) )  =  ( ( F " y )  u.  ( F " { z } ) )
4240, 41syl6eqr 2430 . . . . . . . 8  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  (
( F " y
)  u.  (  _I 
`  ( F " { z } ) ) )  =  ( F " ( y  u.  { z } ) ) )
43 simprr 734 . . . . . . . . 9  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  ( F " y )  ~<_  y )
44 ssdomg 7082 . . . . . . . . . . . 12  |-  ( { ( F `  z
) }  e.  _V  ->  ( ( F " { z } ) 
C_  { ( F `
 z ) }  ->  ( F " { z } )  ~<_  { ( F `  z ) } ) )
4535, 34, 44mpsyl 61 . . . . . . . . . . 11  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  ( F " { z } )  ~<_  { ( F `
 z ) } )
46 fvex 5675 . . . . . . . . . . . . 13  |-  ( F `
 z )  e. 
_V
4746ensn1 7100 . . . . . . . . . . . 12  |-  { ( F `  z ) }  ~~  1o
4831ensn1 7100 . . . . . . . . . . . 12  |-  { z }  ~~  1o
4947, 48entr4i 7093 . . . . . . . . . . 11  |-  { ( F `  z ) }  ~~  { z }
50 domentr 7095 . . . . . . . . . . 11  |-  ( ( ( F " {
z } )  ~<_  { ( F `  z
) }  /\  {
( F `  z
) }  ~~  {
z } )  -> 
( F " {
z } )  ~<_  { z } )
5145, 49, 50sylancl 644 . . . . . . . . . 10  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  ( F " { z } )  ~<_  { z } )
5239, 51eqbrtrd 4166 . . . . . . . . 9  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  (  _I  `  ( F " { z } ) )  ~<_  { z } )
53 simplr 732 . . . . . . . . . 10  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  -.  z  e.  y )
54 disjsn 3804 . . . . . . . . . 10  |-  ( ( y  i^i  { z } )  =  (/)  <->  -.  z  e.  y )
5553, 54sylibr 204 . . . . . . . . 9  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  (
y  i^i  { z } )  =  (/) )
56 undom 7125 . . . . . . . . 9  |-  ( ( ( ( F "
y )  ~<_  y  /\  (  _I  `  ( F
" { z } ) )  ~<_  { z } )  /\  (
y  i^i  { z } )  =  (/) )  ->  ( ( F
" y )  u.  (  _I  `  ( F " { z } ) ) )  ~<_  ( y  u.  { z } ) )
5743, 52, 55, 56syl21anc 1183 . . . . . . . 8  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  (
( F " y
)  u.  (  _I 
`  ( F " { z } ) ) )  ~<_  ( y  u.  { z } ) )
5842, 57eqbrtrrd 4168 . . . . . . 7  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  ( F " ( y  u. 
{ z } ) )  ~<_  ( y  u. 
{ z } ) )
5958exp32 589 . . . . . 6  |-  ( ( y  e.  Fin  /\  -.  z  e.  y
)  ->  ( F  Fn  A  ->  ( ( F " y )  ~<_  y  ->  ( F " ( y  u.  {
z } ) )  ~<_  ( y  u.  {
z } ) ) ) )
6059a2d 24 . . . . 5  |-  ( ( y  e.  Fin  /\  -.  z  e.  y
)  ->  ( ( F  Fn  A  ->  ( F " y )  ~<_  y )  ->  ( F  Fn  A  ->  ( F " ( y  u.  { z } ) )  ~<_  ( y  u.  { z } ) ) ) )
619, 13, 17, 21, 24, 60findcard2s 7278 . . . 4  |-  ( A  e.  Fin  ->  ( F  Fn  A  ->  ( F " A )  ~<_  A ) )
623, 61syl5 30 . . 3  |-  ( A  e.  Fin  ->  ( F : A -onto-> B  -> 
( F " A
)  ~<_  A ) )
6362imp 419 . 2  |-  ( ( A  e.  Fin  /\  F : A -onto-> B )  ->  ( F " A )  ~<_  A )
642, 63eqbrtrrd 4168 1  |-  ( ( A  e.  Fin  /\  F : A -onto-> B )  ->  B  ~<_  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2892    u. cun 3254    i^i cin 3255    C_ wss 3256   (/)c0 3564   {csn 3750   <.cop 3753   class class class wbr 4146    _I cid 4427   ran crn 4812    |` cres 4813   "cima 4814   Fun wfun 5381    Fn wfn 5382   -onto->wfo 5385   ` cfv 5387   1oc1o 6646    ~~ cen 7035    ~<_ cdom 7036   Fincfn 7038
This theorem is referenced by:  fodomfib  7315  fofinf1o  7316  fidomdm  7317  fofi  7321  pwfilem  7329  cmpsub  17378  alexsubALT  17996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-1o 6653  df-er 6834  df-en 7039  df-dom 7040  df-fin 7042
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