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Theorem domssex 7206
Description: Weakening of domssex 7206 to forget the functions in favor of dominance and equinumerosity. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
domssex  |-  ( A  ~<_  B  ->  E. x
( A  C_  x  /\  B  ~~  x ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem domssex
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 brdomi 7057 . 2  |-  ( A  ~<_  B  ->  E. f 
f : A -1-1-> B
)
2 reldom 7053 . . 3  |-  Rel  ~<_
32brrelex2i 4861 . 2  |-  ( A  ~<_  B  ->  B  e.  _V )
4 vex 2904 . . . . . . . 8  |-  f  e. 
_V
5 f1stres 6309 . . . . . . . . . 10  |-  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) : ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) --> ( B  \  ran  f )
65a1i 11 . . . . . . . . 9  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) : ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) --> ( B  \  ran  f ) )
7 difexg 4294 . . . . . . . . . . 11  |-  ( B  e.  _V  ->  ( B  \  ran  f )  e.  _V )
87adantl 453 . . . . . . . . . 10  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  ( B  \  ran  f )  e.  _V )
9 snex 4348 . . . . . . . . . 10  |-  { ~P U.
ran  A }  e.  _V
10 xpexg 4931 . . . . . . . . . 10  |-  ( ( ( B  \  ran  f )  e.  _V  /\ 
{ ~P U. ran  A }  e.  _V )  ->  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
)  e.  _V )
118, 9, 10sylancl 644 . . . . . . . . 9  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  ( ( B 
\  ran  f )  X.  { ~P U. ran  A } )  e.  _V )
12 fex2 5545 . . . . . . . . 9  |-  ( ( ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) : ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) --> ( B  \  ran  f )  /\  (
( B  \  ran  f )  X.  { ~P U. ran  A }
)  e.  _V  /\  ( B  \  ran  f
)  e.  _V )  ->  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) )  e.  _V )
136, 11, 8, 12syl3anc 1184 . . . . . . . 8  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) )  e.  _V )
14 unexg 4652 . . . . . . . 8  |-  ( ( f  e.  _V  /\  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A } ) )  e.  _V )  -> 
( f  u.  ( 1st  |`  ( ( B 
\  ran  f )  X.  { ~P U. ran  A } ) ) )  e.  _V )
154, 13, 14sylancr 645 . . . . . . 7  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  e. 
_V )
16 cnvexg 5347 . . . . . . 7  |-  ( ( f  u.  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  e. 
_V  ->  `' ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  e. 
_V )
1715, 16syl 16 . . . . . 6  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  `' ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  e. 
_V )
18 rnexg 5073 . . . . . 6  |-  ( `' ( f  u.  ( 1st  |`  ( ( B 
\  ran  f )  X.  { ~P U. ran  A } ) ) )  e.  _V  ->  ran  `' ( f  u.  ( 1st  |`  ( ( B 
\  ran  f )  X.  { ~P U. ran  A } ) ) )  e.  _V )
1917, 18syl 16 . . . . 5  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  ran  `' (
f  u.  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  e. 
_V )
20 simpl 444 . . . . . . . 8  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  f : A -1-1-> B )
21 f1dm 5585 . . . . . . . . . 10  |-  ( f : A -1-1-> B  ->  dom  f  =  A
)
224dmex 5074 . . . . . . . . . 10  |-  dom  f  e.  _V
2321, 22syl6eqelr 2478 . . . . . . . . 9  |-  ( f : A -1-1-> B  ->  A  e.  _V )
2423adantr 452 . . . . . . . 8  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  A  e.  _V )
25 simpr 448 . . . . . . . 8  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  B  e.  _V )
26 eqid 2389 . . . . . . . . 9  |-  `' ( f  u.  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  =  `' ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )
2726domss2 7204 . . . . . . . 8  |-  ( ( f : A -1-1-> B  /\  A  e.  _V  /\  B  e.  _V )  ->  ( `' ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) ) : B -1-1-onto-> ran  `' ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  /\  A  C_  ran  `' ( f  u.  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  /\  ( `' ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  o.  f )  =  (  _I  |`  A )
) )
2820, 24, 25, 27syl3anc 1184 . . . . . . 7  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  ( `' ( f  u.  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) ) : B -1-1-onto-> ran  `' ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  /\  A  C_  ran  `' ( f  u.  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  /\  ( `' ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  o.  f )  =  (  _I  |`  A )
) )
2928simp2d 970 . . . . . 6  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  A  C_  ran  `' ( f  u.  ( 1st  |`  ( ( B 
\  ran  f )  X.  { ~P U. ran  A } ) ) ) )
3028simp1d 969 . . . . . . 7  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  `' ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) ) : B -1-1-onto-> ran  `' ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) ) )
31 f1oen3g 7061 . . . . . . 7  |-  ( ( `' ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  e. 
_V  /\  `' (
f  u.  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) ) : B -1-1-onto-> ran  `' ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) ) )  ->  B  ~~  ran  `' ( f  u.  ( 1st  |`  ( ( B 
\  ran  f )  X.  { ~P U. ran  A } ) ) ) )
3217, 30, 31syl2anc 643 . . . . . 6  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  B  ~~  ran  `' ( f  u.  ( 1st  |`  ( ( B 
\  ran  f )  X.  { ~P U. ran  A } ) ) ) )
3329, 32jca 519 . . . . 5  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  ( A  C_  ran  `' ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  /\  B  ~~  ran  `' ( f  u.  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) ) ) )
34 sseq2 3315 . . . . . . 7  |-  ( x  =  ran  `' ( f  u.  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  -> 
( A  C_  x  <->  A 
C_  ran  `' (
f  u.  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) ) ) )
35 breq2 4159 . . . . . . 7  |-  ( x  =  ran  `' ( f  u.  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  -> 
( B  ~~  x  <->  B 
~~  ran  `' (
f  u.  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) ) ) )
3634, 35anbi12d 692 . . . . . 6  |-  ( x  =  ran  `' ( f  u.  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  -> 
( ( A  C_  x  /\  B  ~~  x
)  <->  ( A  C_  ran  `' ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  /\  B  ~~  ran  `' ( f  u.  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) ) ) ) )
3736spcegv 2982 . . . . 5  |-  ( ran  `' ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  e. 
_V  ->  ( ( A 
C_  ran  `' (
f  u.  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  /\  B  ~~  ran  `' ( f  u.  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) ) )  ->  E. x ( A 
C_  x  /\  B  ~~  x ) ) )
3819, 33, 37sylc 58 . . . 4  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  E. x ( A 
C_  x  /\  B  ~~  x ) )
3938ex 424 . . 3  |-  ( f : A -1-1-> B  -> 
( B  e.  _V  ->  E. x ( A 
C_  x  /\  B  ~~  x ) ) )
4039exlimiv 1641 . 2  |-  ( E. f  f : A -1-1-> B  ->  ( B  e. 
_V  ->  E. x ( A 
C_  x  /\  B  ~~  x ) ) )
411, 3, 40sylc 58 1  |-  ( A  ~<_  B  ->  E. x
( A  C_  x  /\  B  ~~  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2901    \ cdif 3262    u. cun 3263    C_ wss 3265   ~Pcpw 3744   {csn 3759   U.cuni 3959   class class class wbr 4155    _I cid 4436    X. cxp 4818   `'ccnv 4819   dom cdm 4820   ran crn 4821    |` cres 4822    o. ccom 4824   -->wf 5392   -1-1->wf1 5393   -1-1-onto->wf1o 5395   1stc1st 6288    ~~ cen 7044    ~<_ cdom 7045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-1st 6290  df-2nd 6291  df-en 7048  df-dom 7049
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