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Theorem fodomfi 7151
Description: An onto function implies dominance of domain over range, for finite sets. Unlike fodom 8165 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 5472 . . 3  |-  ( F : A -onto-> B  -> 
( F " A
)  =  B )
21adantl 452 . 2  |-  ( ( A  e.  Fin  /\  F : A -onto-> B )  ->  ( F " A )  =  B )
3 fofn 5469 . . . 4  |-  ( F : A -onto-> B  ->  F  Fn  A )
4 imaeq2 5024 . . . . . . . 8  |-  ( x  =  (/)  ->  ( F
" x )  =  ( F " (/) ) )
5 ima0 5046 . . . . . . . 8  |-  ( F
" (/) )  =  (/)
64, 5syl6eq 2344 . . . . . . 7  |-  ( x  =  (/)  ->  ( F
" x )  =  (/) )
7 id 19 . . . . . . 7  |-  ( x  =  (/)  ->  x  =  (/) )
86, 7breq12d 4052 . . . . . 6  |-  ( x  =  (/)  ->  ( ( F " x )  ~<_  x  <->  (/)  ~<_  (/) ) )
98imbi2d 307 . . . . 5  |-  ( x  =  (/)  ->  ( ( F  Fn  A  -> 
( F " x
)  ~<_  x )  <->  ( F  Fn  A  ->  (/)  ~<_  (/) ) ) )
10 imaeq2 5024 . . . . . . 7  |-  ( x  =  y  ->  ( F " x )  =  ( F " y
) )
11 id 19 . . . . . . 7  |-  ( x  =  y  ->  x  =  y )
1210, 11breq12d 4052 . . . . . 6  |-  ( x  =  y  ->  (
( F " x
)  ~<_  x  <->  ( F " y )  ~<_  y ) )
1312imbi2d 307 . . . . 5  |-  ( x  =  y  ->  (
( F  Fn  A  ->  ( F " x
)  ~<_  x )  <->  ( F  Fn  A  ->  ( F
" y )  ~<_  y ) ) )
14 imaeq2 5024 . . . . . . 7  |-  ( x  =  ( y  u. 
{ z } )  ->  ( F "
x )  =  ( F " ( y  u.  { z } ) ) )
15 id 19 . . . . . . 7  |-  ( x  =  ( y  u. 
{ z } )  ->  x  =  ( y  u.  { z } ) )
1614, 15breq12d 4052 . . . . . 6  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( F
" x )  ~<_  x  <-> 
( F " (
y  u.  { z } ) )  ~<_  ( y  u.  { z } ) ) )
1716imbi2d 307 . . . . 5  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( F  Fn  A  ->  ( F " x )  ~<_  x )  <->  ( F  Fn  A  ->  ( F "
( y  u.  {
z } ) )  ~<_  ( y  u.  {
z } ) ) ) )
18 imaeq2 5024 . . . . . . 7  |-  ( x  =  A  ->  ( F " x )  =  ( F " A
) )
19 id 19 . . . . . . 7  |-  ( x  =  A  ->  x  =  A )
2018, 19breq12d 4052 . . . . . 6  |-  ( x  =  A  ->  (
( F " x
)  ~<_  x  <->  ( F " A )  ~<_  A ) )
2120imbi2d 307 . . . . 5  |-  ( x  =  A  ->  (
( F  Fn  A  ->  ( F " x
)  ~<_  x )  <->  ( F  Fn  A  ->  ( F
" A )  ~<_  A ) ) )
22 0ex 4166 . . . . . . 7  |-  (/)  e.  _V
23220dom 7007 . . . . . 6  |-  (/)  ~<_  (/)
2423a1i 10 . . . . 5  |-  ( F  Fn  A  ->  (/)  ~<_  (/) )
25 fnfun 5357 . . . . . . . . . . . . . . 15  |-  ( F  Fn  A  ->  Fun  F )
2625ad2antrl 708 . . . . . . . . . . . . . 14  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  Fun  F )
27 funressn 5722 . . . . . . . . . . . . . 14  |-  ( Fun 
F  ->  ( F  |` 
{ z } ) 
C_  { <. z ,  ( F `  z ) >. } )
28 rnss 4923 . . . . . . . . . . . . . 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 4718 . . . . . . . . . . . . 13  |-  ( F
" { z } )  =  ran  ( F  |`  { z } )
31 vex 2804 . . . . . . . . . . . . . . 15  |-  z  e. 
_V
3231rnsnop 5169 . . . . . . . . . . . . . 14  |-  ran  { <. z ,  ( F `
 z ) >. }  =  { ( F `  z ) }
3332eqcomi 2300 . . . . . . . . . . . . 13  |-  { ( F `  z ) }  =  ran  { <. z ,  ( F `
 z ) >. }
3429, 30, 333sstr4g 3232 . . . . . . . . . . . 12  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  ( F " { z } )  C_  { ( F `  z ) } )
35 snex 4232 . . . . . . . . . . . 12  |-  { ( F `  z ) }  e.  _V
36 ssexg 4176 . . . . . . . . . . . 12  |-  ( ( ( F " {
z } )  C_  { ( F `  z
) }  /\  {
( F `  z
) }  e.  _V )  ->  ( F " { z } )  e.  _V )
3734, 35, 36sylancl 643 . . . . . . . . . . 11  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  ( F " { z } )  e.  _V )
38 fvi 5595 . . . . . . . . . . 11  |-  ( ( F " { z } )  e.  _V  ->  (  _I  `  ( F " { z } ) )  =  ( F " { z } ) )
3937, 38syl 15 . . . . . . . . . 10  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  (  _I  `  ( F " { z } ) )  =  ( F
" { z } ) )
4039uneq2d 3342 . . . . . . . . 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 5109 . . . . . . . . 9  |-  ( F
" ( y  u. 
{ z } ) )  =  ( ( F " y )  u.  ( F " { z } ) )
4240, 41syl6eqr 2346 . . . . . . . 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 733 . . . . . . . . 9  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  ( F " y )  ~<_  y )
44 ssdomg 6923 . . . . . . . . . . . 12  |-  ( { ( F `  z
) }  e.  _V  ->  ( ( F " { z } ) 
C_  { ( F `
 z ) }  ->  ( F " { z } )  ~<_  { ( F `  z ) } ) )
4535, 34, 44mpsyl 59 . . . . . . . . . . 11  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  ( F " { z } )  ~<_  { ( F `
 z ) } )
46 fvex 5555 . . . . . . . . . . . . 13  |-  ( F `
 z )  e. 
_V
4746ensn1 6941 . . . . . . . . . . . 12  |-  { ( F `  z ) }  ~~  1o
4831ensn1 6941 . . . . . . . . . . . 12  |-  { z }  ~~  1o
4947, 48entr4i 6934 . . . . . . . . . . 11  |-  { ( F `  z ) }  ~~  { z }
50 domentr 6936 . . . . . . . . . . 11  |-  ( ( ( F " {
z } )  ~<_  { ( F `  z
) }  /\  {
( F `  z
) }  ~~  {
z } )  -> 
( F " {
z } )  ~<_  { z } )
5145, 49, 50sylancl 643 . . . . . . . . . 10  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  ( F " { z } )  ~<_  { z } )
5239, 51eqbrtrd 4059 . . . . . . . . 9  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  (  _I  `  ( F " { z } ) )  ~<_  { z } )
53 simplr 731 . . . . . . . . . 10  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  -.  z  e.  y )
54 disjsn 3706 . . . . . . . . . 10  |-  ( ( y  i^i  { z } )  =  (/)  <->  -.  z  e.  y )
5553, 54sylibr 203 . . . . . . . . 9  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  (
y  i^i  { z } )  =  (/) )
56 undom 6966 . . . . . . . . 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 1181 . . . . . . . 8  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  (
( F " y
)  u.  (  _I 
`  ( F " { z } ) ) )  ~<_  ( y  u.  { z } ) )
5842, 57eqbrtrrd 4061 . . . . . . 7  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( F  Fn  A  /\  ( F " y )  ~<_  y ) )  ->  ( F " ( y  u. 
{ z } ) )  ~<_  ( y  u. 
{ z } ) )
5958exp32 588 . . . . . 6  |-  ( ( y  e.  Fin  /\  -.  z  e.  y
)  ->  ( F  Fn  A  ->  ( ( F " y )  ~<_  y  ->  ( F " ( y  u.  {
z } ) )  ~<_  ( y  u.  {
z } ) ) ) )
6059a2d 23 . . . . 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 7115 . . . 4  |-  ( A  e.  Fin  ->  ( F  Fn  A  ->  ( F " A )  ~<_  A ) )
623, 61syl5 28 . . 3  |-  ( A  e.  Fin  ->  ( F : A -onto-> B  -> 
( F " A
)  ~<_  A ) )
6362imp 418 . 2  |-  ( ( A  e.  Fin  /\  F : A -onto-> B )  ->  ( F " A )  ~<_  A )
642, 63eqbrtrrd 4061 1  |-  ( ( A  e.  Fin  /\  F : A -onto-> B )  ->  B  ~<_  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   <.cop 3656   class class class wbr 4039    _I cid 4320   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265    Fn wfn 5266   -onto->wfo 5269   ` cfv 5271   1oc1o 6488    ~~ cen 6876    ~<_ cdom 6877   Fincfn 6879
This theorem is referenced by:  fodomfib  7152  fofinf1o  7153  fidomdm  7154  fofi  7158  pwfilem  7166  cmpsub  17143  alexsubALT  17761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-fin 6883
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