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Theorem domdifsn 7661
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 7604 . . . . 5  |-  ( A 
~<  B  ->  A  ~<_  B )
2 relsdom 7584 . . . . . . 7  |-  Rel  ~<
32brrelex2i 4896 . . . . . 6  |-  ( A 
~<  B  ->  B  e. 
_V )
4 brdomg 7587 . . . . . 6  |-  ( B  e.  _V  ->  ( A  ~<_  B  <->  E. f 
f : A -1-1-> B
) )
53, 4syl 17 . . . . 5  |-  ( A 
~<  B  ->  ( A  ~<_  B  <->  E. f  f : A -1-1-> B ) )
61, 5mpbid 213 . . . 4  |-  ( A 
~<  B  ->  E. f 
f : A -1-1-> B
)
76adantr 466 . . 3  |-  ( ( A  ~<  B  /\  C  e.  B )  ->  E. f  f : A -1-1-> B )
8 f1f 5796 . . . . . . . 8  |-  ( f : A -1-1-> B  -> 
f : A --> B )
9 frn 5752 . . . . . . . 8  |-  ( f : A --> B  ->  ran  f  C_  B )
108, 9syl 17 . . . . . . 7  |-  ( f : A -1-1-> B  ->  ran  f  C_  B )
1110adantl 467 . . . . . 6  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  ran  f  C_  B )
12 sdomnen 7605 . . . . . . . 8  |-  ( A 
~<  B  ->  -.  A  ~~  B )
1312ad2antrr 730 . . . . . . 7  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  -.  A  ~~  B )
14 vex 3090 . . . . . . . . . . 11  |-  f  e. 
_V
15 dff1o5 5840 . . . . . . . . . . . 12  |-  ( f : A -1-1-onto-> B  <->  ( f : A -1-1-> B  /\  ran  f  =  B ) )
1615biimpri 209 . . . . . . . . . . 11  |-  ( ( f : A -1-1-> B  /\  ran  f  =  B )  ->  f : A
-1-1-onto-> B )
17 f1oen3g 7592 . . . . . . . . . . 11  |-  ( ( f  e.  _V  /\  f : A -1-1-onto-> B )  ->  A  ~~  B )
1814, 16, 17sylancr 667 . . . . . . . . . 10  |-  ( ( f : A -1-1-> B  /\  ran  f  =  B )  ->  A  ~~  B )
1918ex 435 . . . . . . . . 9  |-  ( f : A -1-1-> B  -> 
( ran  f  =  B  ->  A  ~~  B
) )
2019necon3bd 2643 . . . . . . . 8  |-  ( f : A -1-1-> B  -> 
( -.  A  ~~  B  ->  ran  f  =/=  B ) )
2120adantl 467 . . . . . . 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 3861 . . . . . 6  |-  ( ( ran  f  C_  B  /\  ran  f  =/=  B
)  ->  ( B  \  ran  f )  =/=  (/) )
2411, 22, 23syl2anc 665 . . . . 5  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  ( B  \  ran  f )  =/=  (/) )
25 n0 3777 . . . . 5  |-  ( ( B  \  ran  f
)  =/=  (/)  <->  E. x  x  e.  ( B  \  ran  f ) )
2624, 25sylib 199 . . . 4  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  E. x  x  e.  ( B  \  ran  f ) )
272brrelexi 4895 . . . . . . . . 9  |-  ( A 
~<  B  ->  A  e. 
_V )
2827ad2antrr 730 . . . . . . . 8  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  A  e.  _V )
293ad2antrr 730 . . . . . . . . 9  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  B  e.  _V )
30 difexg 4573 . . . . . . . . 9  |-  ( B  e.  _V  ->  ( B  \  { x }
)  e.  _V )
3129, 30syl 17 . . . . . . . 8  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  -> 
( B  \  {
x } )  e. 
_V )
32 eldifn 3594 . . . . . . . . . . . . 13  |-  ( x  e.  ( B  \  ran  f )  ->  -.  x  e.  ran  f )
33 disjsn 4063 . . . . . . . . . . . . 13  |-  ( ( ran  f  i^i  {
x } )  =  (/) 
<->  -.  x  e.  ran  f )
3432, 33sylibr 215 . . . . . . . . . . . 12  |-  ( x  e.  ( B  \  ran  f )  ->  ( ran  f  i^i  { x } )  =  (/) )
3534adantl 467 . . . . . . . . . . 11  |-  ( ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f ) )  ->  ( ran  f  i^i  { x }
)  =  (/) )
3610adantr 466 . . . . . . . . . . . 12  |-  ( ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f ) )  ->  ran  f  C_  B )
37 reldisj 3842 . . . . . . . . . . . 12  |-  ( ran  f  C_  B  ->  ( ( ran  f  i^i 
{ x } )  =  (/)  <->  ran  f  C_  ( B  \  { x }
) ) )
3836, 37syl 17 . . . . . . . . . . 11  |-  ( ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f ) )  ->  ( ( ran  f  i^i  { x } )  =  (/)  <->  ran  f  C_  ( B  \  { x } ) ) )
3935, 38mpbid 213 . . . . . . . . . 10  |-  ( ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f ) )  ->  ran  f  C_  ( B  \  { x } ) )
40 f1ssr 5802 . . . . . . . . . 10  |-  ( ( f : A -1-1-> B  /\  ran  f  C_  ( B  \  { x }
) )  ->  f : A -1-1-> ( B  \  { x } ) )
4139, 40syldan 472 . . . . . . . . 9  |-  ( ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f ) )  ->  f : A -1-1-> ( B  \  { x } ) )
4241adantl 467 . . . . . . . 8  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  -> 
f : A -1-1-> ( B  \  { x } ) )
43 f1dom2g 7594 . . . . . . . 8  |-  ( ( A  e.  _V  /\  ( B  \  { x } )  e.  _V  /\  f : A -1-1-> ( B  \  { x } ) )  ->  A  ~<_  ( B  \  { x } ) )
4428, 31, 42, 43syl3anc 1264 . . . . . . 7  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  A  ~<_  ( B  \  { x } ) )
45 eldifi 3593 . . . . . . . . 9  |-  ( x  e.  ( B  \  ran  f )  ->  x  e.  B )
4645ad2antll 733 . . . . . . . 8  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  x  e.  B )
47 simplr 760 . . . . . . . 8  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  C  e.  B )
48 difsnen 7660 . . . . . . . 8  |-  ( ( B  e.  _V  /\  x  e.  B  /\  C  e.  B )  ->  ( B  \  {
x } )  ~~  ( B  \  { C } ) )
4929, 46, 47, 48syl3anc 1264 . . . . . . 7  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  -> 
( B  \  {
x } )  ~~  ( B  \  { C } ) )
50 domentr 7635 . . . . . . 7  |-  ( ( A  ~<_  ( B  \  { x } )  /\  ( B  \  { x } ) 
~~  ( B  \  { C } ) )  ->  A  ~<_  ( B 
\  { C }
) )
5144, 49, 50syl2anc 665 . . . . . 6  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  ( f : A -1-1-> B  /\  x  e.  ( B  \  ran  f
) ) )  ->  A  ~<_  ( B  \  { C } ) )
5251expr 618 . . . . 5  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  ( x  e.  ( B  \  ran  f )  ->  A  ~<_  ( B  \  { C } ) ) )
5352exlimdv 1771 . . . 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 1773 . 2  |-  ( ( A  ~<  B  /\  C  e.  B )  ->  A  ~<_  ( B  \  { C } ) )
561adantr 466 . . 3  |-  ( ( A  ~<  B  /\  -.  C  e.  B
)  ->  A  ~<_  B )
57 difsn 4137 . . . . 5  |-  ( -.  C  e.  B  -> 
( B  \  { C } )  =  B )
5857breq2d 4438 . . . 4  |-  ( -.  C  e.  B  -> 
( A  ~<_  ( B 
\  { C }
)  <->  A  ~<_  B )
)
5958adantl 467 . . 3  |-  ( ( A  ~<  B  /\  -.  C  e.  B
)  ->  ( A  ~<_  ( B  \  { C } )  <->  A  ~<_  B ) )
6056, 59mpbird 235 . 2  |-  ( ( A  ~<  B  /\  -.  C  e.  B
)  ->  A  ~<_  ( B 
\  { C }
) )
6155, 60pm2.61dan 798 1  |-  ( A 
~<  B  ->  A  ~<_  ( B  \  { C } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1870    =/= wne 2625   _Vcvv 3087    \ cdif 3439    i^i cin 3441    C_ wss 3442   (/)c0 3767   {csn 4002   class class class wbr 4426   ran crn 4855   -->wf 5597   -1-1->wf1 5598   -1-1-onto->wf1o 5600    ~~ cen 7574    ~<_ cdom 7575    ~< csdm 7576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-suc 5448  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-1o 7190  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580
This theorem is referenced by:  domunsn  7728  marypha1lem  7953
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