MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fodomfi Structured version   Unicode version

Theorem fodomfi 7377
Description: An onto function implies dominance of domain over range, for finite sets. Unlike fodom 8392 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 5650 . . 3  |-  ( F : A -onto-> B  -> 
( F " A
)  =  B )
21adantl 453 . 2  |-  ( ( A  e.  Fin  /\  F : A -onto-> B )  ->  ( F " A )  =  B )
3 fofn 5647 . . . 4  |-  ( F : A -onto-> B  ->  F  Fn  A )
4 imaeq2 5191 . . . . . . . 8  |-  ( x  =  (/)  ->  ( F
" x )  =  ( F " (/) ) )
5 ima0 5213 . . . . . . . 8  |-  ( F
" (/) )  =  (/)
64, 5syl6eq 2483 . . . . . . 7  |-  ( x  =  (/)  ->  ( F
" x )  =  (/) )
7 id 20 . . . . . . 7  |-  ( x  =  (/)  ->  x  =  (/) )
86, 7breq12d 4217 . . . . . 6  |-  ( x  =  (/)  ->  ( ( F " x )  ~<_  x  <->  (/)  ~<_  (/) ) )
98imbi2d 308 . . . . 5  |-  ( x  =  (/)  ->  ( ( F  Fn  A  -> 
( F " x
)  ~<_  x )  <->  ( F  Fn  A  ->  (/)  ~<_  (/) ) ) )
10 imaeq2 5191 . . . . . . 7  |-  ( x  =  y  ->  ( F " x )  =  ( F " y
) )
11 id 20 . . . . . . 7  |-  ( x  =  y  ->  x  =  y )
1210, 11breq12d 4217 . . . . . 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 5191 . . . . . . 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 4217 . . . . . 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 5191 . . . . . . 7  |-  ( x  =  A  ->  ( F " x )  =  ( F " A
) )
19 id 20 . . . . . . 7  |-  ( x  =  A  ->  x  =  A )
2018, 19breq12d 4217 . . . . . 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 4331 . . . . . . 7  |-  (/)  e.  _V
23220dom 7229 . . . . . 6  |-  (/)  ~<_  (/)
2423a1i 11 . . . . 5  |-  ( F  Fn  A  ->  (/)  ~<_  (/) )
25 fnfun 5534 . . . . . . . . . . . . . . 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 5911 . . . . . . . . . . . . . 14  |-  ( Fun 
F  ->  ( F  |` 
{ z } ) 
C_  { <. z ,  ( F `  z ) >. } )
28 rnss 5090 . . . . . . . . . . . . . 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 4883 . . . . . . . . . . . . 13  |-  ( F
" { z } )  =  ran  ( F  |`  { z } )
31 vex 2951 . . . . . . . . . . . . . . 15  |-  z  e. 
_V
3231rnsnop 5342 . . . . . . . . . . . . . 14  |-  ran  { <. z ,  ( F `
 z ) >. }  =  { ( F `  z ) }
3332eqcomi 2439 . . . . . . . . . . . . 13  |-  { ( F `  z ) }  =  ran  { <. z ,  ( F `
 z ) >. }
3429, 30, 333sstr4g 3381 . . . . . . . . . . . 12  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  ( F " { z } )  C_  { ( F `  z ) } )
35 snex 4397 . . . . . . . . . . . 12  |-  { ( F `  z ) }  e.  _V
36 ssexg 4341 . . . . . . . . . . . 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 5775 . . . . . . . . . . 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 3493 . . . . . . . . 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 5276 . . . . . . . . 9  |-  ( F
" ( y  u. 
{ z } ) )  =  ( ( F " y )  u.  ( F " { z } ) )
4240, 41syl6eqr 2485 . . . . . . . 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 7145 . . . . . . . . . . . 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 5734 . . . . . . . . . . . . 13  |-  ( F `
 z )  e. 
_V
4746ensn1 7163 . . . . . . . . . . . 12  |-  { ( F `  z ) }  ~~  1o
4831ensn1 7163 . . . . . . . . . . . 12  |-  { z }  ~~  1o
4947, 48entr4i 7156 . . . . . . . . . . 11  |-  { ( F `  z ) }  ~~  { z }
50 domentr 7158 . . . . . . . . . . 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 4224 . . . . . . . . 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 3860 . . . . . . . . . 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 7188 . . . . . . . . 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 4226 . . . . . . 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 7341 . . . 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 4226 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 1652    e. wcel 1725   _Vcvv 2948    u. cun 3310    i^i cin 3311    C_ wss 3312   (/)c0 3620   {csn 3806   <.cop 3809   class class class wbr 4204    _I cid 4485   ran crn 4871    |` cres 4872   "cima 4873   Fun wfun 5440    Fn wfn 5441   -onto->wfo 5444   ` cfv 5446   1oc1o 6709    ~~ cen 7098    ~<_ cdom 7099   Fincfn 7101
This theorem is referenced by:  fodomfib  7378  fofinf1o  7379  fidomdm  7380  fofi  7384  pwfilem  7393  cmpsub  17453  alexsubALT  18072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-1o 6716  df-er 6897  df-en 7102  df-dom 7103  df-fin 7105
  Copyright terms: Public domain W3C validator