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

Theorem domdifsn 7499
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 7442 . . . . 5  |-  ( A 
~<  B  ->  A  ~<_  B )
2 relsdom 7422 . . . . . . 7  |-  Rel  ~<
32brrelex2i 4983 . . . . . 6  |-  ( A 
~<  B  ->  B  e. 
_V )
4 brdomg 7425 . . . . . 6  |-  ( B  e.  _V  ->  ( A  ~<_  B  <->  E. f 
f : A -1-1-> B
) )
53, 4syl 16 . . . . 5  |-  ( A 
~<  B  ->  ( A  ~<_  B  <->  E. f  f : A -1-1-> B ) )
61, 5mpbid 210 . . . 4  |-  ( A 
~<  B  ->  E. f 
f : A -1-1-> B
)
76adantr 465 . . 3  |-  ( ( A  ~<  B  /\  C  e.  B )  ->  E. f  f : A -1-1-> B )
8 f1f 5709 . . . . . . . 8  |-  ( f : A -1-1-> B  -> 
f : A --> B )
9 frn 5668 . . . . . . . 8  |-  ( f : A --> B  ->  ran  f  C_  B )
108, 9syl 16 . . . . . . 7  |-  ( f : A -1-1-> B  ->  ran  f  C_  B )
1110adantl 466 . . . . . 6  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  ran  f  C_  B )
12 sdomnen 7443 . . . . . . . 8  |-  ( A 
~<  B  ->  -.  A  ~~  B )
1312ad2antrr 725 . . . . . . 7  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  -.  A  ~~  B )
14 vex 3075 . . . . . . . . . . 11  |-  f  e. 
_V
15 dff1o5 5753 . . . . . . . . . . . 12  |-  ( f : A -1-1-onto-> B  <->  ( f : A -1-1-> B  /\  ran  f  =  B ) )
1615biimpri 206 . . . . . . . . . . 11  |-  ( ( f : A -1-1-> B  /\  ran  f  =  B )  ->  f : A
-1-1-onto-> B )
17 f1oen3g 7430 . . . . . . . . . . 11  |-  ( ( f  e.  _V  /\  f : A -1-1-onto-> B )  ->  A  ~~  B )
1814, 16, 17sylancr 663 . . . . . . . . . 10  |-  ( ( f : A -1-1-> B  /\  ran  f  =  B )  ->  A  ~~  B )
1918ex 434 . . . . . . . . 9  |-  ( f : A -1-1-> B  -> 
( ran  f  =  B  ->  A  ~~  B
) )
2019necon3bd 2661 . . . . . . . 8  |-  ( f : A -1-1-> B  -> 
( -.  A  ~~  B  ->  ran  f  =/=  B ) )
2120adantl 466 . . . . . . 7  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  ( -.  A  ~~  B  ->  ran  f  =/=  B ) )
2213, 21mpd 15 . . . . . 6  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  ran  f  =/= 
B )
23 pssdifn0 3843 . . . . . 6  |-  ( ( ran  f  C_  B  /\  ran  f  =/=  B
)  ->  ( B  \  ran  f )  =/=  (/) )
2411, 22, 23syl2anc 661 . . . . 5  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  ( B  \  ran  f )  =/=  (/) )
25 n0 3749 . . . . 5  |-  ( ( B  \  ran  f
)  =/=  (/)  <->  E. x  x  e.  ( B  \  ran  f ) )
2624, 25sylib 196 . . . 4  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  E. x  x  e.  ( B  \  ran  f ) )
272brrelexi 4982 . . . . . . . . 9  |-  ( A 
~<  B  ->  A  e. 
_V )
2827ad2antrr 725 . . . . . . . 8  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  A  e.  _V )
293ad2antrr 725 . . . . . . . . 9  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  B  e.  _V )
30 difexg 4543 . . . . . . . . 9  |-  ( B  e.  _V  ->  ( B  \  { x }
)  e.  _V )
3129, 30syl 16 . . . . . . . 8  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  -> 
( B  \  {
x } )  e. 
_V )
32 eldifn 3582 . . . . . . . . . . . . 13  |-  ( x  e.  ( B  \  ran  f )  ->  -.  x  e.  ran  f )
33 disjsn 4039 . . . . . . . . . . . . 13  |-  ( ( ran  f  i^i  {
x } )  =  (/) 
<->  -.  x  e.  ran  f )
3432, 33sylibr 212 . . . . . . . . . . . 12  |-  ( x  e.  ( B  \  ran  f )  ->  ( ran  f  i^i  { x } )  =  (/) )
3534adantl 466 . . . . . . . . . . 11  |-  ( ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f ) )  ->  ( ran  f  i^i  { x }
)  =  (/) )
3610adantr 465 . . . . . . . . . . . 12  |-  ( ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f ) )  ->  ran  f  C_  B )
37 reldisj 3825 . . . . . . . . . . . 12  |-  ( ran  f  C_  B  ->  ( ( ran  f  i^i 
{ x } )  =  (/)  <->  ran  f  C_  ( B  \  { x }
) ) )
3836, 37syl 16 . . . . . . . . . . 11  |-  ( ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f ) )  ->  ( ( ran  f  i^i  { x } )  =  (/)  <->  ran  f  C_  ( B  \  { x } ) ) )
3935, 38mpbid 210 . . . . . . . . . 10  |-  ( ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f ) )  ->  ran  f  C_  ( B  \  { x } ) )
40 f1ssr 5715 . . . . . . . . . 10  |-  ( ( f : A -1-1-> B  /\  ran  f  C_  ( B  \  { x }
) )  ->  f : A -1-1-> ( B  \  { x } ) )
4139, 40syldan 470 . . . . . . . . 9  |-  ( ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f ) )  ->  f : A -1-1-> ( B  \  { x } ) )
4241adantl 466 . . . . . . . 8  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  -> 
f : A -1-1-> ( B  \  { x } ) )
43 f1dom2g 7432 . . . . . . . 8  |-  ( ( A  e.  _V  /\  ( B  \  { x } )  e.  _V  /\  f : A -1-1-> ( B  \  { x } ) )  ->  A  ~<_  ( B  \  { x } ) )
4428, 31, 42, 43syl3anc 1219 . . . . . . 7  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  A  ~<_  ( B  \  { x } ) )
45 eldifi 3581 . . . . . . . . 9  |-  ( x  e.  ( B  \  ran  f )  ->  x  e.  B )
4645ad2antll 728 . . . . . . . 8  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  x  e.  B )
47 simplr 754 . . . . . . . 8  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  C  e.  B )
48 difsnen 7498 . . . . . . . 8  |-  ( ( B  e.  _V  /\  x  e.  B  /\  C  e.  B )  ->  ( B  \  {
x } )  ~~  ( B  \  { C } ) )
4929, 46, 47, 48syl3anc 1219 . . . . . . 7  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  -> 
( B  \  {
x } )  ~~  ( B  \  { C } ) )
50 domentr 7473 . . . . . . 7  |-  ( ( A  ~<_  ( B  \  { x } )  /\  ( B  \  { x } ) 
~~  ( B  \  { C } ) )  ->  A  ~<_  ( B 
\  { C }
) )
5144, 49, 50syl2anc 661 . . . . . 6  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  A  ~<_  ( B  \  { C } ) )
5251expr 615 . . . . 5  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  ( x  e.  ( B  \  ran  f )  ->  A  ~<_  ( B  \  { C } ) ) )
5352exlimdv 1691 . . . 4  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  ( E. x  x  e.  ( B  \  ran  f )  ->  A  ~<_  ( B 
\  { C }
) ) )
5426, 53mpd 15 . . 3  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  A  ~<_  ( B 
\  { C }
) )
557, 54exlimddv 1693 . 2  |-  ( ( A  ~<  B  /\  C  e.  B )  ->  A  ~<_  ( B  \  { C } ) )
561adantr 465 . . 3  |-  ( ( A  ~<  B  /\  -.  C  e.  B
)  ->  A  ~<_  B )
57 difsn 4111 . . . . 5  |-  ( -.  C  e.  B  -> 
( B  \  { C } )  =  B )
5857breq2d 4407 . . . 4  |-  ( -.  C  e.  B  -> 
( A  ~<_  ( B 
\  { C }
)  <->  A  ~<_  B )
)
5958adantl 466 . . 3  |-  ( ( A  ~<  B  /\  -.  C  e.  B
)  ->  ( A  ~<_  ( B  \  { C } )  <->  A  ~<_  B ) )
6056, 59mpbird 232 . 2  |-  ( ( A  ~<  B  /\  -.  C  e.  B
)  ->  A  ~<_  ( B 
\  { C }
) )
6155, 60pm2.61dan 789 1  |-  ( A 
~<  B  ->  A  ~<_  ( B  \  { C } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2645   _Vcvv 3072    \ cdif 3428    i^i cin 3430    C_ wss 3431   (/)c0 3740   {csn 3980   class class class wbr 4395   ran crn 4944   -->wf 5517   -1-1->wf1 5518   -1-1-onto->wf1o 5520    ~~ cen 7412    ~<_ cdom 7413    ~< csdm 7414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-id 4739  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-1o 7025  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418
This theorem is referenced by:  domunsn  7566  marypha1lem  7789
  Copyright terms: Public domain W3C validator