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Theorem domssex 7260
Description: Weakening of domssex 7260 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 7111 . 2  |-  ( A  ~<_  B  ->  E. f 
f : A -1-1-> B
)
2 reldom 7107 . . 3  |-  Rel  ~<_
32brrelex2i 4911 . 2  |-  ( A  ~<_  B  ->  B  e.  _V )
4 vex 2951 . . . . . . . 8  |-  f  e. 
_V
5 f1stres 6360 . . . . . . . . . 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 4343 . . . . . . . . . . 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 4397 . . . . . . . . . 10  |-  { ~P U.
ran  A }  e.  _V
10 xpexg 4981 . . . . . . . . . 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 5595 . . . . . . . . 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 4702 . . . . . . . 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 5397 . . . . . . 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 5123 . . . . . 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 5635 . . . . . . . . . 10  |-  ( f : A -1-1-> B  ->  dom  f  =  A
)
224dmex 5124 . . . . . . . . . 10  |-  dom  f  e.  _V
2321, 22syl6eqelr 2524 . . . . . . . . 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 2435 . . . . . . . . 9  |-  `' ( f  u.  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  =  `' ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )
2726domss2 7258 . . . . . . . 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 7115 . . . . . . 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 3362 . . . . . . 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 4208 . . . . . . 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 3029 . . . . 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 1644 . 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 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948    \ cdif 3309    u. cun 3310    C_ wss 3312   ~Pcpw 3791   {csn 3806   U.cuni 4007   class class class wbr 4204    _I cid 4485    X. cxp 4868   `'ccnv 4869   dom cdm 4870   ran crn 4871    |` cres 4872    o. ccom 4874   -->wf 5442   -1-1->wf1 5443   -1-1-onto->wf1o 5445   1stc1st 6339    ~~ cen 7098    ~<_ cdom 7099
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-1st 6341  df-2nd 6342  df-en 7102  df-dom 7103
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