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Theorem fodomfi 7847
Description: An onto function implies dominance of domain over range, for finite sets. Unlike fodom 8941 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 5806 . . 3  |-  ( F : A -onto-> B  -> 
( F " A
)  =  B )
21adantl 467 . 2  |-  ( ( A  e.  Fin  /\  F : A -onto-> B )  ->  ( F " A )  =  B )
3 fofn 5803 . . . 4  |-  ( F : A -onto-> B  ->  F  Fn  A )
4 imaeq2 5175 . . . . . . . 8  |-  ( x  =  (/)  ->  ( F
" x )  =  ( F " (/) ) )
5 ima0 5194 . . . . . . . 8  |-  ( F
" (/) )  =  (/)
64, 5syl6eq 2477 . . . . . . 7  |-  ( x  =  (/)  ->  ( F
" x )  =  (/) )
7 id 23 . . . . . . 7  |-  ( x  =  (/)  ->  x  =  (/) )
86, 7breq12d 4430 . . . . . 6  |-  ( x  =  (/)  ->  ( ( F " x )  ~<_  x  <->  (/)  ~<_  (/) ) )
98imbi2d 317 . . . . 5  |-  ( x  =  (/)  ->  ( ( F  Fn  A  -> 
( F " x
)  ~<_  x )  <->  ( F  Fn  A  ->  (/)  ~<_  (/) ) ) )
10 imaeq2 5175 . . . . . . 7  |-  ( x  =  y  ->  ( F " x )  =  ( F " y
) )
11 id 23 . . . . . . 7  |-  ( x  =  y  ->  x  =  y )
1210, 11breq12d 4430 . . . . . 6  |-  ( x  =  y  ->  (
( F " x
)  ~<_  x  <->  ( F " y )  ~<_  y ) )
1312imbi2d 317 . . . . 5  |-  ( x  =  y  ->  (
( F  Fn  A  ->  ( F " x
)  ~<_  x )  <->  ( F  Fn  A  ->  ( F
" y )  ~<_  y ) ) )
14 imaeq2 5175 . . . . . . 7  |-  ( x  =  ( y  u. 
{ z } )  ->  ( F "
x )  =  ( F " ( y  u.  { z } ) ) )
15 id 23 . . . . . . 7  |-  ( x  =  ( y  u. 
{ z } )  ->  x  =  ( y  u.  { z } ) )
1614, 15breq12d 4430 . . . . . 6  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( F
" x )  ~<_  x  <-> 
( F " (
y  u.  { z } ) )  ~<_  ( y  u.  { z } ) ) )
1716imbi2d 317 . . . . 5  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( F  Fn  A  ->  ( F " x )  ~<_  x )  <->  ( F  Fn  A  ->  ( F "
( y  u.  {
z } ) )  ~<_  ( y  u.  {
z } ) ) ) )
18 imaeq2 5175 . . . . . . 7  |-  ( x  =  A  ->  ( F " x )  =  ( F " A
) )
19 id 23 . . . . . . 7  |-  ( x  =  A  ->  x  =  A )
2018, 19breq12d 4430 . . . . . 6  |-  ( x  =  A  ->  (
( F " x
)  ~<_  x  <->  ( F " A )  ~<_  A ) )
2120imbi2d 317 . . . . 5  |-  ( x  =  A  ->  (
( F  Fn  A  ->  ( F " x
)  ~<_  x )  <->  ( F  Fn  A  ->  ( F
" A )  ~<_  A ) ) )
22 0ex 4548 . . . . . . 7  |-  (/)  e.  _V
23220dom 7699 . . . . . 6  |-  (/)  ~<_  (/)
2423a1i 11 . . . . 5  |-  ( F  Fn  A  ->  (/)  ~<_  (/) )
25 fnfun 5682 . . . . . . . . . . . . . . 15  |-  ( F  Fn  A  ->  Fun  F )
2625ad2antrl 732 . . . . . . . . . . . . . 14  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  Fun  F )
27 funressn 6083 . . . . . . . . . . . . . 14  |-  ( Fun 
F  ->  ( F  |` 
{ z } ) 
C_  { <. z ,  ( F `  z ) >. } )
28 rnss 5074 . . . . . . . . . . . . . 14  |-  ( ( F  |`  { z } )  C_  { <. z ,  ( F `  z ) >. }  ->  ran  ( F  |`  { z } )  C_  ran  {
<. z ,  ( F `
 z ) >. } )
2926, 27, 283syl 18 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  ran  ( F  |`  { z } )  C_  ran  {
<. z ,  ( F `
 z ) >. } )
30 df-ima 4858 . . . . . . . . . . . . 13  |-  ( F
" { z } )  =  ran  ( F  |`  { z } )
31 vex 3081 . . . . . . . . . . . . . . 15  |-  z  e. 
_V
3231rnsnop 5328 . . . . . . . . . . . . . 14  |-  ran  { <. z ,  ( F `
 z ) >. }  =  { ( F `  z ) }
3332eqcomi 2433 . . . . . . . . . . . . 13  |-  { ( F `  z ) }  =  ran  { <. z ,  ( F `
 z ) >. }
3429, 30, 333sstr4g 3502 . . . . . . . . . . . 12  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  ( F " { z } )  C_  { ( F `  z ) } )
35 snex 4654 . . . . . . . . . . . 12  |-  { ( F `  z ) }  e.  _V
36 ssexg 4562 . . . . . . . . . . . 12  |-  ( ( ( F " {
z } )  C_  { ( F `  z
) }  /\  {
( F `  z
) }  e.  _V )  ->  ( F " { z } )  e.  _V )
3734, 35, 36sylancl 666 . . . . . . . . . . 11  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  ( F " { z } )  e.  _V )
38 fvi 5929 . . . . . . . . . . 11  |-  ( ( F " { z } )  e.  _V  ->  (  _I  `  ( F " { z } ) )  =  ( F " { z } ) )
3937, 38syl 17 . . . . . . . . . 10  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  (  _I  `  ( F " { z } ) )  =  ( F
" { z } ) )
4039uneq2d 3617 . . . . . . . . 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 5259 . . . . . . . . 9  |-  ( F
" ( y  u. 
{ z } ) )  =  ( ( F " y )  u.  ( F " { z } ) )
4240, 41syl6eqr 2479 . . . . . . . 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 764 . . . . . . . . 9  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  ( F " y )  ~<_  y )
44 ssdomg 7613 . . . . . . . . . . . 12  |-  ( { ( F `  z
) }  e.  _V  ->  ( ( F " { z } ) 
C_  { ( F `
 z ) }  ->  ( F " { z } )  ~<_  { ( F `  z ) } ) )
4535, 34, 44mpsyl 65 . . . . . . . . . . 11  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  ( F " { z } )  ~<_  { ( F `
 z ) } )
46 fvex 5882 . . . . . . . . . . . . 13  |-  ( F `
 z )  e. 
_V
4746ensn1 7631 . . . . . . . . . . . 12  |-  { ( F `  z ) }  ~~  1o
4831ensn1 7631 . . . . . . . . . . . 12  |-  { z }  ~~  1o
4947, 48entr4i 7624 . . . . . . . . . . 11  |-  { ( F `  z ) }  ~~  { z }
50 domentr 7626 . . . . . . . . . . 11  |-  ( ( ( F " {
z } )  ~<_  { ( F `  z
) }  /\  {
( F `  z
) }  ~~  {
z } )  -> 
( F " {
z } )  ~<_  { z } )
5145, 49, 50sylancl 666 . . . . . . . . . 10  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  ( F " { z } )  ~<_  { z } )
5239, 51eqbrtrd 4437 . . . . . . . . 9  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  (  _I  `  ( F " { z } ) )  ~<_  { z } )
53 simplr 760 . . . . . . . . . 10  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  -.  z  e.  y )
54 disjsn 4054 . . . . . . . . . 10  |-  ( ( y  i^i  { z } )  =  (/)  <->  -.  z  e.  y )
5553, 54sylibr 215 . . . . . . . . 9  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  (
y  i^i  { z } )  =  (/) )
56 undom 7657 . . . . . . . . 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 1263 . . . . . . . 8  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  (
( F " y
)  u.  (  _I 
`  ( F " { z } ) ) )  ~<_  ( y  u.  { z } ) )
5842, 57eqbrtrrd 4439 . . . . . . 7  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  ( F " ( y  u. 
{ z } ) )  ~<_  ( y  u. 
{ z } ) )
5958exp32 608 . . . . . 6  |-  ( ( y  e.  Fin  /\  -.  z  e.  y
)  ->  ( F  Fn  A  ->  ( ( F " y )  ~<_  y  ->  ( F " ( y  u.  {
z } ) )  ~<_  ( y  u.  {
z } ) ) ) )
6059a2d 29 . . . . 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 7809 . . . 4  |-  ( A  e.  Fin  ->  ( F  Fn  A  ->  ( F " A )  ~<_  A ) )
623, 61syl5 33 . . 3  |-  ( A  e.  Fin  ->  ( F : A -onto-> B  -> 
( F " A
)  ~<_  A ) )
6362imp 430 . 2  |-  ( ( A  e.  Fin  /\  F : A -onto-> B )  ->  ( F " A )  ~<_  A )
642, 63eqbrtrrd 4439 1  |-  ( ( A  e.  Fin  /\  F : A -onto-> B )  ->  B  ~<_  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867   _Vcvv 3078    u. cun 3431    i^i cin 3432    C_ wss 3433   (/)c0 3758   {csn 3993   <.cop 3999   class class class wbr 4417    _I cid 4755   ran crn 4846    |` cres 4847   "cima 4848   Fun wfun 5586    Fn wfn 5587   -onto->wfo 5590   ` cfv 5592   1oc1o 7174    ~~ cen 7565    ~<_ cdom 7566   Fincfn 7568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-om 6698  df-1o 7181  df-er 7362  df-en 7569  df-dom 7570  df-fin 7572
This theorem is referenced by:  fodomfib  7848  fofinf1o  7849  fidomdm  7850  fofi  7857  pwfilem  7865  cmpsub  20352  alexsubALT  21003  phpreu  31677  poimirlem26  31714
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