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

Theorem domdifsn 7593
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 7536 . . . . 5  |-  ( A 
~<  B  ->  A  ~<_  B )
2 relsdom 7516 . . . . . . 7  |-  Rel  ~<
32brrelex2i 5030 . . . . . 6  |-  ( A 
~<  B  ->  B  e. 
_V )
4 brdomg 7519 . . . . . 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 463 . . 3  |-  ( ( A  ~<  B  /\  C  e.  B )  ->  E. f  f : A -1-1-> B )
8 f1f 5763 . . . . . . . 8  |-  ( f : A -1-1-> B  -> 
f : A --> B )
9 frn 5719 . . . . . . . 8  |-  ( f : A --> B  ->  ran  f  C_  B )
108, 9syl 16 . . . . . . 7  |-  ( f : A -1-1-> B  ->  ran  f  C_  B )
1110adantl 464 . . . . . 6  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  ran  f  C_  B )
12 sdomnen 7537 . . . . . . . 8  |-  ( A 
~<  B  ->  -.  A  ~~  B )
1312ad2antrr 723 . . . . . . 7  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  -.  A  ~~  B )
14 vex 3109 . . . . . . . . . . 11  |-  f  e. 
_V
15 dff1o5 5807 . . . . . . . . . . . 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 7524 . . . . . . . . . . 11  |-  ( ( f  e.  _V  /\  f : A -1-1-onto-> B )  ->  A  ~~  B )
1814, 16, 17sylancr 661 . . . . . . . . . 10  |-  ( ( f : A -1-1-> B  /\  ran  f  =  B )  ->  A  ~~  B )
1918ex 432 . . . . . . . . 9  |-  ( f : A -1-1-> B  -> 
( ran  f  =  B  ->  A  ~~  B
) )
2019necon3bd 2666 . . . . . . . 8  |-  ( f : A -1-1-> B  -> 
( -.  A  ~~  B  ->  ran  f  =/=  B ) )
2120adantl 464 . . . . . . 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 3876 . . . . . 6  |-  ( ( ran  f  C_  B  /\  ran  f  =/=  B
)  ->  ( B  \  ran  f )  =/=  (/) )
2411, 22, 23syl2anc 659 . . . . 5  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  ( B  \  ran  f )  =/=  (/) )
25 n0 3793 . . . . 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 5029 . . . . . . . . 9  |-  ( A 
~<  B  ->  A  e. 
_V )
2827ad2antrr 723 . . . . . . . 8  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  A  e.  _V )
293ad2antrr 723 . . . . . . . . 9  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  B  e.  _V )
30 difexg 4585 . . . . . . . . 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 3613 . . . . . . . . . . . . 13  |-  ( x  e.  ( B  \  ran  f )  ->  -.  x  e.  ran  f )
33 disjsn 4076 . . . . . . . . . . . . 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 464 . . . . . . . . . . 11  |-  ( ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f ) )  ->  ( ran  f  i^i  { x }
)  =  (/) )
3610adantr 463 . . . . . . . . . . . 12  |-  ( ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f ) )  ->  ran  f  C_  B )
37 reldisj 3858 . . . . . . . . . . . 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 5769 . . . . . . . . . 10  |-  ( ( f : A -1-1-> B  /\  ran  f  C_  ( B  \  { x }
) )  ->  f : A -1-1-> ( B  \  { x } ) )
4139, 40syldan 468 . . . . . . . . 9  |-  ( ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f ) )  ->  f : A -1-1-> ( B  \  { x } ) )
4241adantl 464 . . . . . . . 8  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  -> 
f : A -1-1-> ( B  \  { x } ) )
43 f1dom2g 7526 . . . . . . . 8  |-  ( ( A  e.  _V  /\  ( B  \  { x } )  e.  _V  /\  f : A -1-1-> ( B  \  { x } ) )  ->  A  ~<_  ( B  \  { x } ) )
4428, 31, 42, 43syl3anc 1226 . . . . . . 7  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  A  ~<_  ( B  \  { x } ) )
45 eldifi 3612 . . . . . . . . 9  |-  ( x  e.  ( B  \  ran  f )  ->  x  e.  B )
4645ad2antll 726 . . . . . . . 8  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  x  e.  B )
47 simplr 753 . . . . . . . 8  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  C  e.  B )
48 difsnen 7592 . . . . . . . 8  |-  ( ( B  e.  _V  /\  x  e.  B  /\  C  e.  B )  ->  ( B  \  {
x } )  ~~  ( B  \  { C } ) )
4929, 46, 47, 48syl3anc 1226 . . . . . . 7  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  -> 
( B  \  {
x } )  ~~  ( B  \  { C } ) )
50 domentr 7567 . . . . . . 7  |-  ( ( A  ~<_  ( B  \  { x } )  /\  ( B  \  { x } ) 
~~  ( B  \  { C } ) )  ->  A  ~<_  ( B 
\  { C }
) )
5144, 49, 50syl2anc 659 . . . . . 6  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  A  ~<_  ( B  \  { C } ) )
5251expr 613 . . . . 5  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  ( x  e.  ( B  \  ran  f )  ->  A  ~<_  ( B  \  { C } ) ) )
5352exlimdv 1729 . . . 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 1731 . 2  |-  ( ( A  ~<  B  /\  C  e.  B )  ->  A  ~<_  ( B  \  { C } ) )
561adantr 463 . . 3  |-  ( ( A  ~<  B  /\  -.  C  e.  B
)  ->  A  ~<_  B )
57 difsn 4150 . . . . 5  |-  ( -.  C  e.  B  -> 
( B  \  { C } )  =  B )
5857breq2d 4451 . . . 4  |-  ( -.  C  e.  B  -> 
( A  ~<_  ( B 
\  { C }
)  <->  A  ~<_  B )
)
5958adantl 464 . . 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 367    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649   _Vcvv 3106    \ cdif 3458    i^i cin 3460    C_ wss 3461   (/)c0 3783   {csn 4016   class class class wbr 4439   ran crn 4989   -->wf 5566   -1-1->wf1 5567   -1-1-onto->wf1o 5569    ~~ cen 7506    ~<_ cdom 7507    ~< csdm 7508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-1o 7122  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512
This theorem is referenced by:  domunsn  7660  marypha1lem  7885
  Copyright terms: Public domain W3C validator