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Theorem infdifsn 7357
Description: Removing a singleton from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
infdifsn  |-  ( om  ~<_  A  ->  ( A  \  { B } ) 
~~  A )

Proof of Theorem infdifsn
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 brdomi 6873 . . . 4  |-  ( om  ~<_  A  ->  E. f 
f : om -1-1-> A
)
21adantr 451 . . 3  |-  ( ( om  ~<_  A  /\  B  e.  A )  ->  E. f 
f : om -1-1-> A
)
3 reldom 6869 . . . . . . . . 9  |-  Rel  ~<_
43brrelex2i 4730 . . . . . . . 8  |-  ( om  ~<_  A  ->  A  e.  _V )
54ad2antrr 706 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  A  e.  _V )
6 simplr 731 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  B  e.  A )
7 f1f 5437 . . . . . . . . 9  |-  ( f : om -1-1-> A  -> 
f : om --> A )
87adantl 452 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  f : om
--> A )
9 peano1 4675 . . . . . . . 8  |-  (/)  e.  om
10 ffvelrn 5663 . . . . . . . 8  |-  ( ( f : om --> A  /\  (/) 
e.  om )  ->  (
f `  (/) )  e.  A )
118, 9, 10sylancl 643 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( f `  (/) )  e.  A
)
12 difsnen 6944 . . . . . . 7  |-  ( ( A  e.  _V  /\  B  e.  A  /\  ( f `  (/) )  e.  A )  ->  ( A  \  { B }
)  ~~  ( A  \  { ( f `  (/) ) } ) )
135, 6, 11, 12syl3anc 1182 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( A  \  { B } ) 
~~  ( A  \  { ( f `  (/) ) } ) )
14 vex 2791 . . . . . . . . . . . 12  |-  f  e. 
_V
15 f1f1orn 5483 . . . . . . . . . . . . 13  |-  ( f : om -1-1-> A  -> 
f : om -1-1-onto-> ran  f )
1615adantl 452 . . . . . . . . . . . 12  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  f : om
-1-1-onto-> ran  f )
17 f1oen3g 6877 . . . . . . . . . . . 12  |-  ( ( f  e.  _V  /\  f : om -1-1-onto-> ran  f )  ->  om  ~~  ran  f )
1814, 16, 17sylancr 644 . . . . . . . . . . 11  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  om  ~~  ran  f )
19 ensym 6910 . . . . . . . . . . 11  |-  ( om 
~~  ran  f  ->  ran  f  ~~  om )
2018, 19syl 15 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ran  f  ~~  om )
213brrelexi 4729 . . . . . . . . . . . . 13  |-  ( om  ~<_  A  ->  om  e.  _V )
2221ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  om  e.  _V )
23 limom 4671 . . . . . . . . . . . . 13  |-  Lim  om
2423limenpsi 7036 . . . . . . . . . . . 12  |-  ( om  e.  _V  ->  om  ~~  ( om  \  { (/) } ) )
2522, 24syl 15 . . . . . . . . . . 11  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  om  ~~  ( om  \  { (/) } ) )
2614resex 4995 . . . . . . . . . . . . 13  |-  ( f  |`  ( om  \  { (/)
} ) )  e. 
_V
27 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  f : om
-1-1-> A )
28 difss 3303 . . . . . . . . . . . . . 14  |-  ( om 
\  { (/) } ) 
C_  om
29 f1ores 5487 . . . . . . . . . . . . . 14  |-  ( ( f : om -1-1-> A  /\  ( om  \  { (/)
} )  C_  om )  ->  ( f  |`  ( om  \  { (/) } ) ) : ( om 
\  { (/) } ) -1-1-onto-> ( f " ( om 
\  { (/) } ) ) )
3027, 28, 29sylancl 643 . . . . . . . . . . . . 13  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( f  |`  ( om  \  { (/)
} ) ) : ( om  \  { (/)
} ) -1-1-onto-> ( f " ( om  \  { (/) } ) ) )
31 f1oen3g 6877 . . . . . . . . . . . . 13  |-  ( ( ( f  |`  ( om  \  { (/) } ) )  e.  _V  /\  ( f  |`  ( om  \  { (/) } ) ) : ( om 
\  { (/) } ) -1-1-onto-> ( f " ( om 
\  { (/) } ) ) )  ->  ( om  \  { (/) } ) 
~~  ( f "
( om  \  { (/)
} ) ) )
3226, 30, 31sylancr 644 . . . . . . . . . . . 12  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( om  \  { (/) } )  ~~  ( f " ( om  \  { (/) } ) ) )
33 f1orn 5482 . . . . . . . . . . . . . . 15  |-  ( f : om -1-1-onto-> ran  f  <->  ( f  Fn  om  /\  Fun  `' f ) )
3433simprbi 450 . . . . . . . . . . . . . 14  |-  ( f : om -1-1-onto-> ran  f  ->  Fun  `' f )
35 imadif 5327 . . . . . . . . . . . . . 14  |-  ( Fun  `' f  ->  ( f
" ( om  \  { (/)
} ) )  =  ( ( f " om )  \  (
f " { (/) } ) ) )
3616, 34, 353syl 18 . . . . . . . . . . . . 13  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( f " ( om  \  { (/)
} ) )  =  ( ( f " om )  \  (
f " { (/) } ) ) )
37 f1fn 5438 . . . . . . . . . . . . . . . 16  |-  ( f : om -1-1-> A  -> 
f  Fn  om )
3837adantl 452 . . . . . . . . . . . . . . 15  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  f  Fn  om )
39 fnima 5362 . . . . . . . . . . . . . . 15  |-  ( f  Fn  om  ->  (
f " om )  =  ran  f )
4038, 39syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( f " om )  =  ran  f )
41 fnsnfv 5582 . . . . . . . . . . . . . . . 16  |-  ( ( f  Fn  om  /\  (/) 
e.  om )  ->  { ( f `  (/) ) }  =  ( f " { (/) } ) )
4238, 9, 41sylancl 643 . . . . . . . . . . . . . . 15  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  { (
f `  (/) ) }  =  ( f " { (/) } ) )
4342eqcomd 2288 . . . . . . . . . . . . . 14  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( f " { (/) } )  =  { ( f `  (/) ) } )
4440, 43difeq12d 3295 . . . . . . . . . . . . 13  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( (
f " om )  \  ( f " { (/) } ) )  =  ( ran  f  \  { ( f `  (/) ) } ) )
4536, 44eqtrd 2315 . . . . . . . . . . . 12  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( f " ( om  \  { (/)
} ) )  =  ( ran  f  \  { ( f `  (/) ) } ) )
4632, 45breqtrd 4047 . . . . . . . . . . 11  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( om  \  { (/) } )  ~~  ( ran  f  \  {
( f `  (/) ) } ) )
47 entr 6913 . . . . . . . . . . 11  |-  ( ( om  ~~  ( om 
\  { (/) } )  /\  ( om  \  { (/)
} )  ~~  ( ran  f  \  { ( f `  (/) ) } ) )  ->  om  ~~  ( ran  f  \  {
( f `  (/) ) } ) )
4825, 46, 47syl2anc 642 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  om  ~~  ( ran  f  \  { ( f `  (/) ) } ) )
49 entr 6913 . . . . . . . . . 10  |-  ( ( ran  f  ~~  om  /\ 
om  ~~  ( ran  f  \  { ( f `
 (/) ) } ) )  ->  ran  f  ~~  ( ran  f  \  {
( f `  (/) ) } ) )
5020, 48, 49syl2anc 642 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ran  f  ~~  ( ran  f  \  {
( f `  (/) ) } ) )
51 difexg 4162 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  \  ran  f )  e.  _V )
52 enrefg 6893 . . . . . . . . . 10  |-  ( ( A  \  ran  f
)  e.  _V  ->  ( A  \  ran  f
)  ~~  ( A  \  ran  f ) )
535, 51, 523syl 18 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( A  \  ran  f )  ~~  ( A  \  ran  f
) )
54 disjdif 3526 . . . . . . . . . 10  |-  ( ran  f  i^i  ( A 
\  ran  f )
)  =  (/)
5554a1i 10 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( ran  f  i^i  ( A  \  ran  f ) )  =  (/) )
56 difss 3303 . . . . . . . . . . . 12  |-  ( ran  f  \  { ( f `  (/) ) } )  C_  ran  f
57 ssrin 3394 . . . . . . . . . . . 12  |-  ( ( ran  f  \  {
( f `  (/) ) } )  C_  ran  f  -> 
( ( ran  f  \  { ( f `  (/) ) } )  i^i  ( A  \  ran  f ) )  C_  ( ran  f  i^i  ( A  \  ran  f ) ) )
5856, 57ax-mp 8 . . . . . . . . . . 11  |-  ( ( ran  f  \  {
( f `  (/) ) } )  i^i  ( A 
\  ran  f )
)  C_  ( ran  f  i^i  ( A  \  ran  f ) )
59 sseq0 3486 . . . . . . . . . . 11  |-  ( ( ( ( ran  f  \  { ( f `  (/) ) } )  i^i  ( A  \  ran  f ) )  C_  ( ran  f  i^i  ( A  \  ran  f ) )  /\  ( ran  f  i^i  ( A 
\  ran  f )
)  =  (/) )  -> 
( ( ran  f  \  { ( f `  (/) ) } )  i^i  ( A  \  ran  f ) )  =  (/) )
6058, 54, 59mp2an 653 . . . . . . . . . 10  |-  ( ( ran  f  \  {
( f `  (/) ) } )  i^i  ( A 
\  ran  f )
)  =  (/)
6160a1i 10 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( ( ran  f  \  { ( f `  (/) ) } )  i^i  ( A 
\  ran  f )
)  =  (/) )
62 unen 6943 . . . . . . . . 9  |-  ( ( ( ran  f  ~~  ( ran  f  \  {
( f `  (/) ) } )  /\  ( A 
\  ran  f )  ~~  ( A  \  ran  f ) )  /\  ( ( ran  f  i^i  ( A  \  ran  f ) )  =  (/)  /\  ( ( ran  f  \  { ( f `  (/) ) } )  i^i  ( A 
\  ran  f )
)  =  (/) ) )  ->  ( ran  f  u.  ( A  \  ran  f ) )  ~~  ( ( ran  f  \  { ( f `  (/) ) } )  u.  ( A  \  ran  f ) ) )
6350, 53, 55, 61, 62syl22anc 1183 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( ran  f  u.  ( A  \  ran  f ) ) 
~~  ( ( ran  f  \  { ( f `  (/) ) } )  u.  ( A 
\  ran  f )
) )
64 frn 5395 . . . . . . . . . 10  |-  ( f : om --> A  ->  ran  f  C_  A )
658, 64syl 15 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ran  f  C_  A )
66 undif 3534 . . . . . . . . 9  |-  ( ran  f  C_  A  <->  ( ran  f  u.  ( A  \  ran  f ) )  =  A )
6765, 66sylib 188 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( ran  f  u.  ( A  \  ran  f ) )  =  A )
68 uncom 3319 . . . . . . . . 9  |-  ( ( ran  f  \  {
( f `  (/) ) } )  u.  ( A 
\  ran  f )
)  =  ( ( A  \  ran  f
)  u.  ( ran  f  \  { ( f `  (/) ) } ) )
69 eldifn 3299 . . . . . . . . . . . . 13  |-  ( ( f `  (/) )  e.  ( A  \  ran  f )  ->  -.  ( f `  (/) )  e. 
ran  f )
70 fnfvelrn 5662 . . . . . . . . . . . . . 14  |-  ( ( f  Fn  om  /\  (/) 
e.  om )  ->  (
f `  (/) )  e. 
ran  f )
7138, 9, 70sylancl 643 . . . . . . . . . . . . 13  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( f `  (/) )  e.  ran  f )
7269, 71nsyl3 111 . . . . . . . . . . . 12  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  -.  (
f `  (/) )  e.  ( A  \  ran  f ) )
73 disjsn 3693 . . . . . . . . . . . 12  |-  ( ( ( A  \  ran  f )  i^i  {
( f `  (/) ) } )  =  (/)  <->  -.  (
f `  (/) )  e.  ( A  \  ran  f ) )
7472, 73sylibr 203 . . . . . . . . . . 11  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( ( A  \  ran  f )  i^i  { ( f `
 (/) ) } )  =  (/) )
75 undif4 3511 . . . . . . . . . . 11  |-  ( ( ( A  \  ran  f )  i^i  {
( f `  (/) ) } )  =  (/)  ->  (
( A  \  ran  f )  u.  ( ran  f  \  { ( f `  (/) ) } ) )  =  ( ( ( A  \  ran  f )  u.  ran  f )  \  {
( f `  (/) ) } ) )
7674, 75syl 15 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( ( A  \  ran  f )  u.  ( ran  f  \  { ( f `  (/) ) } ) )  =  ( ( ( A  \  ran  f
)  u.  ran  f
)  \  { (
f `  (/) ) } ) )
77 uncom 3319 . . . . . . . . . . . 12  |-  ( ( A  \  ran  f
)  u.  ran  f
)  =  ( ran  f  u.  ( A 
\  ran  f )
)
7877, 67syl5eq 2327 . . . . . . . . . . 11  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( ( A  \  ran  f )  u.  ran  f )  =  A )
7978difeq1d 3293 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( (
( A  \  ran  f )  u.  ran  f )  \  {
( f `  (/) ) } )  =  ( A 
\  { ( f `
 (/) ) } ) )
8076, 79eqtrd 2315 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( ( A  \  ran  f )  u.  ( ran  f  \  { ( f `  (/) ) } ) )  =  ( A  \  { ( f `  (/) ) } ) )
8168, 80syl5eq 2327 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( ( ran  f  \  { ( f `  (/) ) } )  u.  ( A 
\  ran  f )
)  =  ( A 
\  { ( f `
 (/) ) } ) )
8263, 67, 813brtr3d 4052 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  A  ~~  ( A  \  { ( f `  (/) ) } ) )
83 ensym 6910 . . . . . . 7  |-  ( A 
~~  ( A  \  { ( f `  (/) ) } )  -> 
( A  \  {
( f `  (/) ) } )  ~~  A )
8482, 83syl 15 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( A  \  { ( f `  (/) ) } )  ~~  A )
85 entr 6913 . . . . . 6  |-  ( ( ( A  \  { B } )  ~~  ( A  \  { ( f `
 (/) ) } )  /\  ( A  \  { ( f `  (/) ) } )  ~~  A )  ->  ( A  \  { B }
)  ~~  A )
8613, 84, 85syl2anc 642 . . . . 5  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( A  \  { B } ) 
~~  A )
8786ex 423 . . . 4  |-  ( ( om  ~<_  A  /\  B  e.  A )  ->  (
f : om -1-1-> A  ->  ( A  \  { B } )  ~~  A
) )
8887exlimdv 1664 . . 3  |-  ( ( om  ~<_  A  /\  B  e.  A )  ->  ( E. f  f : om
-1-1-> A  ->  ( A  \  { B } ) 
~~  A ) )
892, 88mpd 14 . 2  |-  ( ( om  ~<_  A  /\  B  e.  A )  ->  ( A  \  { B }
)  ~~  A )
90 difsn 3755 . . . 4  |-  ( -.  B  e.  A  -> 
( A  \  { B } )  =  A )
9190adantl 452 . . 3  |-  ( ( om  ~<_  A  /\  -.  B  e.  A )  ->  ( A  \  { B } )  =  A )
92 enrefg 6893 . . . . 5  |-  ( A  e.  _V  ->  A  ~~  A )
934, 92syl 15 . . . 4  |-  ( om  ~<_  A  ->  A  ~~  A )
9493adantr 451 . . 3  |-  ( ( om  ~<_  A  /\  -.  B  e.  A )  ->  A  ~~  A )
9591, 94eqbrtrd 4043 . 2  |-  ( ( om  ~<_  A  /\  -.  B  e.  A )  ->  ( A  \  { B } )  ~~  A
)
9689, 95pm2.61dan 766 1  |-  ( om  ~<_  A  ->  ( A  \  { B } ) 
~~  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   class class class wbr 4023   omcom 4656   `'ccnv 4688   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249    Fn wfn 5250   -->wf 5251   -1-1->wf1 5252   -1-1-onto->wf1o 5254   ` cfv 5255    ~~ cen 6860    ~<_ cdom 6861
This theorem is referenced by:  infdiffi  7358  infcda1  7819  infpss  7843
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-3or 935  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-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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-1o 6479  df-er 6660  df-en 6864  df-dom 6865
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