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Theorem domdifsn 6945
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 6889 . . . . 5  |-  ( A 
~<  B  ->  A  ~<_  B )
2 relsdom 6870 . . . . . . 7  |-  Rel  ~<
32brrelex2i 4730 . . . . . 6  |-  ( A 
~<  B  ->  B  e. 
_V )
4 brdomg 6872 . . . . . 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 5437 . . . . . . . . . 10  |-  ( f : A -1-1-> B  -> 
f : A --> B )
9 frn 5395 . . . . . . . . . 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 6890 . . . . . . . . . 10  |-  ( A 
~<  B  ->  -.  A  ~~  B )
1312ad2antrr 706 . . . . . . . . 9  |-  ( ( ( A  ~<  B  /\  C  e.  B )  /\  f : A -1-1-> B
)  ->  -.  A  ~~  B )
14 vex 2791 . . . . . . . . . . . . 13  |-  f  e. 
_V
15 dff1o5 5481 . . . . . . . . . . . . . 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 6877 . . . . . . . . . . . . 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 2483 . . . . . . . . . 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 3515 . . . . . . . 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 3464 . . . . . . 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 4729 . . . . . . . . . . 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 4162 . . . . . . . . . . 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 3299 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( B  \  ran  f )  ->  -.  x  e.  ran  f )
33 disjsn 3693 . . . . . . . . . . . . . . 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 3498 . . . . . . . . . . . . . 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 5443 . . . . . . . . . . . 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 6879 . . . . . . . . . 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 3298 . . . . . . . . . . 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 6944 . . . . . . . . . 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 6920 . . . . . . . . 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 1664 . . . . . 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 1664 . . 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 3755 . . . . 5  |-  ( -.  C  e.  B  -> 
( B  \  { C } )  =  B )
6059breq2d 4035 . . . 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 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   class class class wbr 4023   ran crn 4690   -->wf 5251   -1-1->wf1 5252   -1-1-onto->wf1o 5254    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862
This theorem is referenced by:  domunsn  7011  marypha1lem  7186
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866
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