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

Theorem domdifsn 6961
Description: Dominance over a set with one element removed. (Contributed by Stefan O'Rear, 19-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
domdifsn  |-  ( A 
~<  B  ->  A  ~<_  ( B  \  { C } ) )

Proof of Theorem domdifsn
Dummy variables  f  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sdomdom 6905 . . . . 5  |-  ( A 
~<  B  ->  A  ~<_  B )
2 relsdom 6886 . . . . . . 7  |-  Rel  ~<
32brrelex2i 4746 . . . . . 6  |-  ( A 
~<  B  ->  B  e. 
_V )
4 brdomg 6888 . . . . . 6  |-  ( B  e.  _V  ->  ( A  ~<_  B  <->  E. f 
f : A -1-1-> B
) )
53, 4syl 15 . . . . 5  |-  ( A 
~<  B  ->  ( A  ~<_  B  <->  E. f  f : A -1-1-> B ) )
61, 5mpbid 201 . . . 4  |-  ( A 
~<  B  ->  E. f 
f : A -1-1-> B
)
76adantr 451 . . 3  |-  ( ( A  ~<  B  /\  C  e.  B )  ->  E. f  f : A -1-1-> B )
8 f1f 5453 . . . . . . . . . 10  |-  ( f : A -1-1-> B  -> 
f : A --> B )
9 frn 5411 . . . . . . . . . 10  |-  ( f : A --> B  ->  ran  f  C_  B )
108, 9syl 15 . . . . . . . . 9  |-  ( f : A -1-1-> B  ->  ran  f  C_  B )
1110adantl 452 . . . . . . . 8  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  ran  f  C_  B )
12 sdomnen 6906 . . . . . . . . . 10  |-  ( A 
~<  B  ->  -.  A  ~~  B )
1312ad2antrr 706 . . . . . . . . 9  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  -.  A  ~~  B )
14 vex 2804 . . . . . . . . . . . . 13  |-  f  e. 
_V
15 dff1o5 5497 . . . . . . . . . . . . . 14  |-  ( f : A -1-1-onto-> B  <->  ( f : A -1-1-> B  /\  ran  f  =  B ) )
1615biimpri 197 . . . . . . . . . . . . 13  |-  ( ( f : A -1-1-> B  /\  ran  f  =  B )  ->  f : A
-1-1-onto-> B )
17 f1oen3g 6893 . . . . . . . . . . . . 13  |-  ( ( f  e.  _V  /\  f : A -1-1-onto-> B )  ->  A  ~~  B )
1814, 16, 17sylancr 644 . . . . . . . . . . . 12  |-  ( ( f : A -1-1-> B  /\  ran  f  =  B )  ->  A  ~~  B )
1918ex 423 . . . . . . . . . . 11  |-  ( f : A -1-1-> B  -> 
( ran  f  =  B  ->  A  ~~  B
) )
2019necon3bd 2496 . . . . . . . . . 10  |-  ( f : A -1-1-> B  -> 
( -.  A  ~~  B  ->  ran  f  =/=  B ) )
2120adantl 452 . . . . . . . . 9  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  ( -.  A  ~~  B  ->  ran  f  =/=  B ) )
2213, 21mpd 14 . . . . . . . 8  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  ran  f  =/= 
B )
23 pssdifn0 3528 . . . . . . . 8  |-  ( ( ran  f  C_  B  /\  ran  f  =/=  B
)  ->  ( B  \  ran  f )  =/=  (/) )
2411, 22, 23syl2anc 642 . . . . . . 7  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  ( B  \  ran  f )  =/=  (/) )
25 n0 3477 . . . . . . 7  |-  ( ( B  \  ran  f
)  =/=  (/)  <->  E. x  x  e.  ( B  \  ran  f ) )
2624, 25sylib 188 . . . . . 6  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  E. x  x  e.  ( B  \  ran  f ) )
272brrelexi 4745 . . . . . . . . . . 11  |-  ( A 
~<  B  ->  A  e. 
_V )
2827ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  A  e.  _V )
293ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  B  e.  _V )
30 difexg 4178 . . . . . . . . . . 11  |-  ( B  e.  _V  ->  ( B  \  { x }
)  e.  _V )
3129, 30syl 15 . . . . . . . . . 10  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  -> 
( B  \  {
x } )  e. 
_V )
32 eldifn 3312 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( B  \  ran  f )  ->  -.  x  e.  ran  f )
33 disjsn 3706 . . . . . . . . . . . . . . 15  |-  ( ( ran  f  i^i  {
x } )  =  (/) 
<->  -.  x  e.  ran  f )
3432, 33sylibr 203 . . . . . . . . . . . . . 14  |-  ( x  e.  ( B  \  ran  f )  ->  ( ran  f  i^i  { x } )  =  (/) )
3534adantl 452 . . . . . . . . . . . . 13  |-  ( ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f ) )  ->  ( ran  f  i^i  { x }
)  =  (/) )
3610adantr 451 . . . . . . . . . . . . . 14  |-  ( ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f ) )  ->  ran  f  C_  B )
37 reldisj 3511 . . . . . . . . . . . . . 14  |-  ( ran  f  C_  B  ->  ( ( ran  f  i^i 
{ x } )  =  (/)  <->  ran  f  C_  ( B  \  { x }
) ) )
3836, 37syl 15 . . . . . . . . . . . . 13  |-  ( ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f ) )  ->  ( ( ran  f  i^i  { x } )  =  (/)  <->  ran  f  C_  ( B  \  { x } ) ) )
3935, 38mpbid 201 . . . . . . . . . . . 12  |-  ( ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f ) )  ->  ran  f  C_  ( B  \  { x } ) )
40 f1ssr 5459 . . . . . . . . . . . 12  |-  ( ( f : A -1-1-> B  /\  ran  f  C_  ( B  \  { x }
) )  ->  f : A -1-1-> ( B  \  { x } ) )
4139, 40syldan 456 . . . . . . . . . . 11  |-  ( ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f ) )  ->  f : A -1-1-> ( B  \  { x } ) )
4241adantl 452 . . . . . . . . . 10  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  -> 
f : A -1-1-> ( B  \  { x } ) )
43 f1dom2g 6895 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  ( B  \  { x } )  e.  _V  /\  f : A -1-1-> ( B  \  { x } ) )  ->  A  ~<_  ( B  \  { x } ) )
4428, 31, 42, 43syl3anc 1182 . . . . . . . . 9  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  A  ~<_  ( B  \  { x } ) )
45 eldifi 3311 . . . . . . . . . . 11  |-  ( x  e.  ( B  \  ran  f )  ->  x  e.  B )
4645ad2antll 709 . . . . . . . . . 10  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  x  e.  B )
47 simplr 731 . . . . . . . . . 10  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  C  e.  B )
48 difsnen 6960 . . . . . . . . . 10  |-  ( ( B  e.  _V  /\  x  e.  B  /\  C  e.  B )  ->  ( B  \  {
x } )  ~~  ( B  \  { C } ) )
4929, 46, 47, 48syl3anc 1182 . . . . . . . . 9  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  -> 
( B  \  {
x } )  ~~  ( B  \  { C } ) )
50 domentr 6936 . . . . . . . . 9  |-  ( ( A  ~<_  ( B  \  { x } )  /\  ( B  \  { x } ) 
~~  ( B  \  { C } ) )  ->  A  ~<_  ( B 
\  { C }
) )
5144, 49, 50syl2anc 642 . . . . . . . 8  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  A  ~<_  ( B  \  { C } ) )
5251expr 598 . . . . . . 7  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  ( x  e.  ( B  \  ran  f )  ->  A  ~<_  ( B  \  { C } ) ) )
5352exlimdv 1626 . . . . . 6  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  ( E. x  x  e.  ( B  \  ran  f )  ->  A  ~<_  ( B 
\  { C }
) ) )
5426, 53mpd 14 . . . . 5  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  A  ~<_  ( B 
\  { C }
) )
5554ex 423 . . . 4  |-  ( ( A  ~<  B  /\  C  e.  B )  ->  ( f : A -1-1-> B  ->  A  ~<_  ( B 
\  { C }
) ) )
5655exlimdv 1626 . . 3  |-  ( ( A  ~<  B  /\  C  e.  B )  ->  ( E. f  f : A -1-1-> B  ->  A  ~<_  ( B  \  { C } ) ) )
577, 56mpd 14 . 2  |-  ( ( A  ~<  B  /\  C  e.  B )  ->  A  ~<_  ( B  \  { C } ) )
581adantr 451 . . 3  |-  ( ( A  ~<  B  /\  -.  C  e.  B
)  ->  A  ~<_  B )
59 difsn 3768 . . . . 5  |-  ( -.  C  e.  B  -> 
( B  \  { C } )  =  B )
6059breq2d 4051 . . . 4  |-  ( -.  C  e.  B  -> 
( A  ~<_  ( B 
\  { C }
)  <->  A  ~<_  B )
)
6160adantl 452 . . 3  |-  ( ( A  ~<  B  /\  -.  C  e.  B
)  ->  ( A  ~<_  ( B  \  { C } )  <->  A  ~<_  B ) )
6258, 61mpbird 223 . 2  |-  ( ( A  ~<  B  /\  -.  C  e.  B
)  ->  A  ~<_  ( B 
\  { C }
) )
6357, 62pm2.61dan 766 1  |-  ( A 
~<  B  ->  A  ~<_  ( B  \  { C } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   class class class wbr 4039   ran crn 4706   -->wf 5267   -1-1->wf1 5268   -1-1-onto->wf1o 5270    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878
This theorem is referenced by:  domunsn  7027  marypha1lem  7202
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-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-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-suc 4414  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-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882
  Copyright terms: Public domain W3C validator