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Theorem domssex 7022
Description: Weakening of domssex 7022 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 6873 . 2  |-  ( A  ~<_  B  ->  E. f 
f : A -1-1-> B
)
2 reldom 6869 . . 3  |-  Rel  ~<_
32brrelex2i 4730 . 2  |-  ( A  ~<_  B  ->  B  e.  _V )
4 vex 2791 . . . . . . . 8  |-  f  e. 
_V
5 f1stres 6141 . . . . . . . . . 10  |-  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) : ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) --> ( B  \  ran  f )
65a1i 10 . . . . . . . . 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 4162 . . . . . . . . . . 11  |-  ( B  e.  _V  ->  ( B  \  ran  f )  e.  _V )
87adantl 452 . . . . . . . . . 10  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  ( B  \  ran  f )  e.  _V )
9 snex 4216 . . . . . . . . . 10  |-  { ~P U.
ran  A }  e.  _V
10 xpexg 4800 . . . . . . . . . 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 643 . . . . . . . . 9  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  ( ( B 
\  ran  f )  X.  { ~P U. ran  A } )  e.  _V )
12 fex2 5401 . . . . . . . . 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 1182 . . . . . . . 8  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) )  e.  _V )
14 unexg 4521 . . . . . . . 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 644 . . . . . . 7  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  e. 
_V )
16 cnvexg 5208 . . . . . . 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 15 . . . . . 6  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  `' ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  e. 
_V )
18 rnexg 4940 . . . . . 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 15 . . . . 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 443 . . . . . . . 8  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  f : A -1-1-> B )
21 f1dm 5441 . . . . . . . . . 10  |-  ( f : A -1-1-> B  ->  dom  f  =  A
)
224dmex 4941 . . . . . . . . . 10  |-  dom  f  e.  _V
2321, 22syl6eqelr 2372 . . . . . . . . 9  |-  ( f : A -1-1-> B  ->  A  e.  _V )
2423adantr 451 . . . . . . . 8  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  A  e.  _V )
25 simpr 447 . . . . . . . 8  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  B  e.  _V )
26 eqid 2283 . . . . . . . . 9  |-  `' ( f  u.  ( 1st  |`  ( ( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )  =  `' ( f  u.  ( 1st  |`  (
( B  \  ran  f )  X.  { ~P U. ran  A }
) ) )
2726domss2 7020 . . . . . . . 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 1182 . . . . . . 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 968 . . . . . 6  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  A  C_  ran  `' ( f  u.  ( 1st  |`  ( ( B 
\  ran  f )  X.  { ~P U. ran  A } ) ) ) )
3028simp1d 967 . . . . . . 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 6877 . . . . . . 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 642 . . . . . 6  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  B  ~~  ran  `' ( f  u.  ( 1st  |`  ( ( B 
\  ran  f )  X.  { ~P U. ran  A } ) ) ) )
3329, 32jca 518 . . . . 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 3200 . . . . . . 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 4027 . . . . . . 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 691 . . . . . 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 2869 . . . . 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 56 . . . 4  |-  ( ( f : A -1-1-> B  /\  B  e.  _V )  ->  E. x ( A 
C_  x  /\  B  ~~  x ) )
3938ex 423 . . 3  |-  ( f : A -1-1-> B  -> 
( B  e.  _V  ->  E. x ( A 
C_  x  /\  B  ~~  x ) ) )
4039exlimiv 1666 . 2  |-  ( E. f  f : A -1-1-> B  ->  ( B  e. 
_V  ->  E. x ( A 
C_  x  /\  B  ~~  x ) ) )
411, 3, 40sylc 56 1  |-  ( A  ~<_  B  ->  E. x
( A  C_  x  /\  B  ~~  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    u. cun 3150    C_ wss 3152   ~Pcpw 3625   {csn 3640   U.cuni 3827   class class class wbr 4023    _I cid 4304    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691    o. ccom 4693   -->wf 5251   -1-1->wf1 5252   -1-1-onto->wf1o 5254   1stc1st 6120    ~~ cen 6860    ~<_ cdom 6861
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-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6122  df-2nd 6123  df-en 6864  df-dom 6865
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