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Theorem domssex 7670
Description: Weakening of domssex 7670 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 7519 . 2  |-  ( A  ~<_  B  ->  E. f 
f : A -1-1-> B
)
2 reldom 7514 . . 3  |-  Rel  ~<_
32brrelex2i 5035 . 2  |-  ( A  ~<_  B  ->  B  e.  _V )
4 vex 3111 . . . . . . . 8  |-  f  e. 
_V
5 f1stres 6798 . . . . . . . . . 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 4590 . . . . . . . . . . 11  |-  ( B  e.  _V  ->  ( B  \  ran  f )  e.  _V )
87adantl 466 . . . . . . . . . 10  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  ( B  \  ran  f )  e.  _V )
9 snex 4683 . . . . . . . . . 10  |-  { ~P U.
ran  A }  e.  _V
10 xpexg 6704 . . . . . . . . . 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 662 . . . . . . . . 9  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  ( ( B 
\  ran  f )  X.  { ~P U. ran  A } )  e.  _V )
12 fex2 6731 . . . . . . . . 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 1223 . . . . . . . 8  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) )  e.  _V )
14 unexg 6578 . . . . . . . 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 663 . . . . . . 7  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  e. 
_V )
16 cnvexg 6722 . . . . . . 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 6708 . . . . . 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 457 . . . . . . . 8  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  f : A -1-1-> B )
21 f1dm 5778 . . . . . . . . . 10  |-  ( f : A -1-1-> B  ->  dom  f  =  A
)
224dmex 6709 . . . . . . . . . 10  |-  dom  f  e.  _V
2321, 22syl6eqelr 2559 . . . . . . . . 9  |-  ( f : A -1-1-> B  ->  A  e.  _V )
2423adantr 465 . . . . . . . 8  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  A  e.  _V )
25 simpr 461 . . . . . . . 8  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  B  e.  _V )
26 eqid 2462 . . . . . . . . 9  |-  `' ( f  u.  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  =  `' ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )
2726domss2 7668 . . . . . . . 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 1223 . . . . . . 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 1004 . . . . . 6  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  A  C_  ran  `' ( f  u.  ( 1st  |`  ( ( B 
\  ran  f )  X.  { ~P U. ran  A } ) ) ) )
3028simp1d 1003 . . . . . . 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 7523 . . . . . . 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 661 . . . . . 6  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  B  ~~  ran  `' ( f  u.  ( 1st  |`  ( ( B 
\  ran  f )  X.  { ~P U. ran  A } ) ) ) )
3329, 32jca 532 . . . . 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 3521 . . . . . . 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 4446 . . . . . . 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 710 . . . . . 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 3194 . . . . 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 60 . . . 4  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  E. x ( A 
C_  x  /\  B  ~~  x ) )
3938ex 434 . . 3  |-  ( f : A -1-1-> B  -> 
( B  e.  _V  ->  E. x ( A 
C_  x  /\  B  ~~  x ) ) )
4039exlimiv 1693 . 2  |-  ( E. f  f : A -1-1-> B  ->  ( B  e. 
_V  ->  E. x ( A 
C_  x  /\  B  ~~  x ) ) )
411, 3, 40sylc 60 1  |-  ( A  ~<_  B  ->  E. x
( A  C_  x  /\  B  ~~  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374   E.wex 1591    e. wcel 1762   _Vcvv 3108    \ cdif 3468    u. cun 3469    C_ wss 3471   ~Pcpw 4005   {csn 4022   U.cuni 4240   class class class wbr 4442    _I cid 4785    X. cxp 4992   `'ccnv 4993   dom cdm 4994   ran crn 4995    |` cres 4996    o. ccom 4998   -->wf 5577   -1-1->wf1 5578   -1-1-onto->wf1o 5580   1stc1st 6774    ~~ cen 7505    ~<_ cdom 7506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-1st 6776  df-2nd 6777  df-en 7509  df-dom 7510
This theorem is referenced by: (None)
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