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Theorem domssex 7733
Description: Weakening of domssex 7733 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 7580 . 2  |-  ( A  ~<_  B  ->  E. f 
f : A -1-1-> B
)
2 reldom 7575 . . 3  |-  Rel  ~<_
32brrelex2i 4876 . 2  |-  ( A  ~<_  B  ->  B  e.  _V )
4 vex 3048 . . . . . . . 8  |-  f  e. 
_V
5 f1stres 6815 . . . . . . . . . 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 4551 . . . . . . . . . . 11  |-  ( B  e.  _V  ->  ( B  \  ran  f )  e.  _V )
87adantl 468 . . . . . . . . . 10  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  ( B  \  ran  f )  e.  _V )
9 snex 4641 . . . . . . . . . 10  |-  { ~P U.
ran  A }  e.  _V
10 xpexg 6593 . . . . . . . . . 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 668 . . . . . . . . 9  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  ( ( B 
\  ran  f )  X.  { ~P U. ran  A } )  e.  _V )
12 fex2 6748 . . . . . . . . 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 1268 . . . . . . . 8  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) )  e.  _V )
14 unexg 6592 . . . . . . . 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 669 . . . . . . 7  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  e. 
_V )
16 cnvexg 6739 . . . . . . 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 17 . . . . . 6  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  `' ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  e. 
_V )
18 rnexg 6725 . . . . . 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 17 . . . . 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 459 . . . . . . . 8  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  f : A -1-1-> B )
21 f1dm 5783 . . . . . . . . . 10  |-  ( f : A -1-1-> B  ->  dom  f  =  A
)
224dmex 6726 . . . . . . . . . 10  |-  dom  f  e.  _V
2321, 22syl6eqelr 2538 . . . . . . . . 9  |-  ( f : A -1-1-> B  ->  A  e.  _V )
2423adantr 467 . . . . . . . 8  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  A  e.  _V )
25 simpr 463 . . . . . . . 8  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  B  e.  _V )
26 eqid 2451 . . . . . . . . 9  |-  `' ( f  u.  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  =  `' ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )
2726domss2 7731 . . . . . . . 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 1268 . . . . . . 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 1021 . . . . . 6  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  A  C_  ran  `' ( f  u.  ( 1st  |`  ( ( B 
\  ran  f )  X.  { ~P U. ran  A } ) ) ) )
3028simp1d 1020 . . . . . . 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 7585 . . . . . . 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 667 . . . . . 6  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  B  ~~  ran  `' ( f  u.  ( 1st  |`  ( ( B 
\  ran  f )  X.  { ~P U. ran  A } ) ) ) )
3329, 32jca 535 . . . . 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 3454 . . . . . . 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 4406 . . . . . . 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 717 . . . . . 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 3135 . . . . 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 62 . . . 4  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  E. x ( A 
C_  x  /\  B  ~~  x ) )
3938ex 436 . . 3  |-  ( f : A -1-1-> B  -> 
( B  e.  _V  ->  E. x ( A 
C_  x  /\  B  ~~  x ) ) )
4039exlimiv 1776 . 2  |-  ( E. f  f : A -1-1-> B  ->  ( B  e. 
_V  ->  E. x ( A 
C_  x  /\  B  ~~  x ) ) )
411, 3, 40sylc 62 1  |-  ( A  ~<_  B  ->  E. x
( A  C_  x  /\  B  ~~  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444   E.wex 1663    e. wcel 1887   _Vcvv 3045    \ cdif 3401    u. cun 3402    C_ wss 3404   ~Pcpw 3951   {csn 3968   U.cuni 4198   class class class wbr 4402    _I cid 4744    X. cxp 4832   `'ccnv 4833   dom cdm 4834   ran crn 4835    |` cres 4836    o. ccom 4838   -->wf 5578   -1-1->wf1 5579   -1-1-onto->wf1o 5581   1stc1st 6791    ~~ cen 7566    ~<_ cdom 7567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-1st 6793  df-2nd 6794  df-en 7570  df-dom 7571
This theorem is referenced by: (None)
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