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Theorem infdifsn 7567
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 7078 . . . 4  |-  ( om  ~<_  A  ->  E. f 
f : om -1-1-> A
)
21adantr 452 . . 3  |-  ( ( om  ~<_  A  /\  B  e.  A )  ->  E. f 
f : om -1-1-> A
)
3 reldom 7074 . . . . . . 7  |-  Rel  ~<_
43brrelex2i 4878 . . . . . 6  |-  ( om  ~<_  A  ->  A  e.  _V )
54ad2antrr 707 . . . . 5  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  A  e.  _V )
6 simplr 732 . . . . 5  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  B  e.  A )
7 f1f 5598 . . . . . . 7  |-  ( f : om -1-1-> A  -> 
f : om --> A )
87adantl 453 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  f : om
--> A )
9 peano1 4823 . . . . . 6  |-  (/)  e.  om
10 ffvelrn 5827 . . . . . 6  |-  ( ( f : om --> A  /\  (/) 
e.  om )  ->  (
f `  (/) )  e.  A )
118, 9, 10sylancl 644 . . . . 5  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( f `  (/) )  e.  A
)
12 difsnen 7149 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  A  /\  ( f `  (/) )  e.  A )  ->  ( A  \  { B }
)  ~~  ( A  \  { ( f `  (/) ) } ) )
135, 6, 11, 12syl3anc 1184 . . . 4  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( A  \  { B } ) 
~~  ( A  \  { ( f `  (/) ) } ) )
14 vex 2919 . . . . . . . . . 10  |-  f  e. 
_V
15 f1f1orn 5644 . . . . . . . . . . 11  |-  ( f : om -1-1-> A  -> 
f : om -1-1-onto-> ran  f )
1615adantl 453 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  f : om
-1-1-onto-> ran  f )
17 f1oen3g 7082 . . . . . . . . . 10  |-  ( ( f  e.  _V  /\  f : om -1-1-onto-> ran  f )  ->  om  ~~  ran  f )
1814, 16, 17sylancr 645 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  om  ~~  ran  f )
1918ensymd 7117 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ran  f  ~~  om )
203brrelexi 4877 . . . . . . . . . . 11  |-  ( om  ~<_  A  ->  om  e.  _V )
2120ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  om  e.  _V )
22 limom 4819 . . . . . . . . . . 11  |-  Lim  om
2322limenpsi 7241 . . . . . . . . . 10  |-  ( om  e.  _V  ->  om  ~~  ( om  \  { (/) } ) )
2421, 23syl 16 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  om  ~~  ( om  \  { (/) } ) )
2514resex 5145 . . . . . . . . . . 11  |-  ( f  |`  ( om  \  { (/)
} ) )  e. 
_V
26 simpr 448 . . . . . . . . . . . 12  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  f : om
-1-1-> A )
27 difss 3434 . . . . . . . . . . . 12  |-  ( om 
\  { (/) } ) 
C_  om
28 f1ores 5648 . . . . . . . . . . . 12  |-  ( ( f : om -1-1-> A  /\  ( om  \  { (/)
} )  C_  om )  ->  ( f  |`  ( om  \  { (/) } ) ) : ( om 
\  { (/) } ) -1-1-onto-> ( f " ( om 
\  { (/) } ) ) )
2926, 27, 28sylancl 644 . . . . . . . . . . 11  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( f  |`  ( om  \  { (/)
} ) ) : ( om  \  { (/)
} ) -1-1-onto-> ( f " ( om  \  { (/) } ) ) )
30 f1oen3g 7082 . . . . . . . . . . 11  |-  ( ( ( f  |`  ( om  \  { (/) } ) )  e.  _V  /\  ( f  |`  ( om  \  { (/) } ) ) : ( om 
\  { (/) } ) -1-1-onto-> ( f " ( om 
\  { (/) } ) ) )  ->  ( om  \  { (/) } ) 
~~  ( f "
( om  \  { (/)
} ) ) )
3125, 29, 30sylancr 645 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( om  \  { (/) } )  ~~  ( f " ( om  \  { (/) } ) ) )
32 f1orn 5643 . . . . . . . . . . . . 13  |-  ( f : om -1-1-onto-> ran  f  <->  ( f  Fn  om  /\  Fun  `' f ) )
3332simprbi 451 . . . . . . . . . . . 12  |-  ( f : om -1-1-onto-> ran  f  ->  Fun  `' f )
34 imadif 5487 . . . . . . . . . . . 12  |-  ( Fun  `' f  ->  ( f
" ( om  \  { (/)
} ) )  =  ( ( f " om )  \  (
f " { (/) } ) ) )
3516, 33, 343syl 19 . . . . . . . . . . 11  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( f " ( om  \  { (/)
} ) )  =  ( ( f " om )  \  (
f " { (/) } ) ) )
36 f1fn 5599 . . . . . . . . . . . . . 14  |-  ( f : om -1-1-> A  -> 
f  Fn  om )
3736adantl 453 . . . . . . . . . . . . 13  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  f  Fn  om )
38 fnima 5522 . . . . . . . . . . . . 13  |-  ( f  Fn  om  ->  (
f " om )  =  ran  f )
3937, 38syl 16 . . . . . . . . . . . 12  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( f " om )  =  ran  f )
40 fnsnfv 5745 . . . . . . . . . . . . . 14  |-  ( ( f  Fn  om  /\  (/) 
e.  om )  ->  { ( f `  (/) ) }  =  ( f " { (/) } ) )
4137, 9, 40sylancl 644 . . . . . . . . . . . . 13  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  { (
f `  (/) ) }  =  ( f " { (/) } ) )
4241eqcomd 2409 . . . . . . . . . . . 12  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( f " { (/) } )  =  { ( f `  (/) ) } )
4339, 42difeq12d 3426 . . . . . . . . . . 11  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( (
f " om )  \  ( f " { (/) } ) )  =  ( ran  f  \  { ( f `  (/) ) } ) )
4435, 43eqtrd 2436 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( f " ( om  \  { (/)
} ) )  =  ( ran  f  \  { ( f `  (/) ) } ) )
4531, 44breqtrd 4196 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( om  \  { (/) } )  ~~  ( ran  f  \  {
( f `  (/) ) } ) )
46 entr 7118 . . . . . . . . 9  |-  ( ( om  ~~  ( om 
\  { (/) } )  /\  ( om  \  { (/)
} )  ~~  ( ran  f  \  { ( f `  (/) ) } ) )  ->  om  ~~  ( ran  f  \  {
( f `  (/) ) } ) )
4724, 45, 46syl2anc 643 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  om  ~~  ( ran  f  \  { ( f `  (/) ) } ) )
48 entr 7118 . . . . . . . 8  |-  ( ( ran  f  ~~  om  /\ 
om  ~~  ( ran  f  \  { ( f `
 (/) ) } ) )  ->  ran  f  ~~  ( ran  f  \  {
( f `  (/) ) } ) )
4919, 47, 48syl2anc 643 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ran  f  ~~  ( ran  f  \  {
( f `  (/) ) } ) )
50 difexg 4311 . . . . . . . 8  |-  ( A  e.  _V  ->  ( A  \  ran  f )  e.  _V )
51 enrefg 7098 . . . . . . . 8  |-  ( ( A  \  ran  f
)  e.  _V  ->  ( A  \  ran  f
)  ~~  ( A  \  ran  f ) )
525, 50, 513syl 19 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( A  \  ran  f )  ~~  ( A  \  ran  f
) )
53 disjdif 3660 . . . . . . . 8  |-  ( ran  f  i^i  ( A 
\  ran  f )
)  =  (/)
5453a1i 11 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( ran  f  i^i  ( A  \  ran  f ) )  =  (/) )
55 difss 3434 . . . . . . . . . 10  |-  ( ran  f  \  { ( f `  (/) ) } )  C_  ran  f
56 ssrin 3526 . . . . . . . . . 10  |-  ( ( ran  f  \  {
( f `  (/) ) } )  C_  ran  f  -> 
( ( ran  f  \  { ( f `  (/) ) } )  i^i  ( A  \  ran  f ) )  C_  ( ran  f  i^i  ( A  \  ran  f ) ) )
5755, 56ax-mp 8 . . . . . . . . 9  |-  ( ( ran  f  \  {
( f `  (/) ) } )  i^i  ( A 
\  ran  f )
)  C_  ( ran  f  i^i  ( A  \  ran  f ) )
58 sseq0 3619 . . . . . . . . 9  |-  ( ( ( ( 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 ) )  =  (/) )
5957, 53, 58mp2an 654 . . . . . . . 8  |-  ( ( ran  f  \  {
( f `  (/) ) } )  i^i  ( A 
\  ran  f )
)  =  (/)
6059a1i 11 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( ( ran  f  \  { ( f `  (/) ) } )  i^i  ( A 
\  ran  f )
)  =  (/) )
61 unen 7148 . . . . . . 7  |-  ( ( ( 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 ) ) )
6249, 52, 54, 60, 61syl22anc 1185 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( ran  f  u.  ( A  \  ran  f ) ) 
~~  ( ( ran  f  \  { ( f `  (/) ) } )  u.  ( A 
\  ran  f )
) )
63 frn 5556 . . . . . . . 8  |-  ( f : om --> A  ->  ran  f  C_  A )
648, 63syl 16 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ran  f  C_  A )
65 undif 3668 . . . . . . 7  |-  ( ran  f  C_  A  <->  ( ran  f  u.  ( A  \  ran  f ) )  =  A )
6664, 65sylib 189 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( ran  f  u.  ( A  \  ran  f ) )  =  A )
67 uncom 3451 . . . . . . 7  |-  ( ( ran  f  \  {
( f `  (/) ) } )  u.  ( A 
\  ran  f )
)  =  ( ( A  \  ran  f
)  u.  ( ran  f  \  { ( f `  (/) ) } ) )
68 eldifn 3430 . . . . . . . . . . 11  |-  ( ( f `  (/) )  e.  ( A  \  ran  f )  ->  -.  ( f `  (/) )  e. 
ran  f )
69 fnfvelrn 5826 . . . . . . . . . . . 12  |-  ( ( f  Fn  om  /\  (/) 
e.  om )  ->  (
f `  (/) )  e. 
ran  f )
7037, 9, 69sylancl 644 . . . . . . . . . . 11  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( f `  (/) )  e.  ran  f )
7168, 70nsyl3 113 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  -.  (
f `  (/) )  e.  ( A  \  ran  f ) )
72 disjsn 3828 . . . . . . . . . 10  |-  ( ( ( A  \  ran  f )  i^i  {
( f `  (/) ) } )  =  (/)  <->  -.  (
f `  (/) )  e.  ( A  \  ran  f ) )
7371, 72sylibr 204 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( ( A  \  ran  f )  i^i  { ( f `
 (/) ) } )  =  (/) )
74 undif4 3644 . . . . . . . . 9  |-  ( ( ( A  \  ran  f )  i^i  {
( f `  (/) ) } )  =  (/)  ->  (
( A  \  ran  f )  u.  ( ran  f  \  { ( f `  (/) ) } ) )  =  ( ( ( A  \  ran  f )  u.  ran  f )  \  {
( f `  (/) ) } ) )
7573, 74syl 16 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( ( A  \  ran  f )  u.  ( ran  f  \  { ( f `  (/) ) } ) )  =  ( ( ( A  \  ran  f
)  u.  ran  f
)  \  { (
f `  (/) ) } ) )
76 uncom 3451 . . . . . . . . . 10  |-  ( ( A  \  ran  f
)  u.  ran  f
)  =  ( ran  f  u.  ( A 
\  ran  f )
)
7776, 66syl5eq 2448 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( ( A  \  ran  f )  u.  ran  f )  =  A )
7877difeq1d 3424 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( (
( A  \  ran  f )  u.  ran  f )  \  {
( f `  (/) ) } )  =  ( A 
\  { ( f `
 (/) ) } ) )
7975, 78eqtrd 2436 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( ( A  \  ran  f )  u.  ( ran  f  \  { ( f `  (/) ) } ) )  =  ( A  \  { ( f `  (/) ) } ) )
8067, 79syl5eq 2448 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( ( ran  f  \  { ( f `  (/) ) } )  u.  ( A 
\  ran  f )
)  =  ( A 
\  { ( f `
 (/) ) } ) )
8162, 66, 803brtr3d 4201 . . . . 5  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  A  ~~  ( A  \  { ( f `  (/) ) } ) )
8281ensymd 7117 . . . 4  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( A  \  { ( f `  (/) ) } )  ~~  A )
83 entr 7118 . . . 4  |-  ( ( ( A  \  { B } )  ~~  ( A  \  { ( f `
 (/) ) } )  /\  ( A  \  { ( f `  (/) ) } )  ~~  A )  ->  ( A  \  { B }
)  ~~  A )
8413, 82, 83syl2anc 643 . . 3  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( A  \  { B } ) 
~~  A )
852, 84exlimddv 1645 . 2  |-  ( ( om  ~<_  A  /\  B  e.  A )  ->  ( A  \  { B }
)  ~~  A )
86 difsn 3893 . . . 4  |-  ( -.  B  e.  A  -> 
( A  \  { B } )  =  A )
8786adantl 453 . . 3  |-  ( ( om  ~<_  A  /\  -.  B  e.  A )  ->  ( A  \  { B } )  =  A )
88 enrefg 7098 . . . . 5  |-  ( A  e.  _V  ->  A  ~~  A )
894, 88syl 16 . . . 4  |-  ( om  ~<_  A  ->  A  ~~  A )
9089adantr 452 . . 3  |-  ( ( om  ~<_  A  /\  -.  B  e.  A )  ->  A  ~~  A )
9187, 90eqbrtrd 4192 . 2  |-  ( ( om  ~<_  A  /\  -.  B  e.  A )  ->  ( A  \  { B } )  ~~  A
)
9285, 91pm2.61dan 767 1  |-  ( om  ~<_  A  ->  ( A  \  { B } ) 
~~  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   _Vcvv 2916    \ cdif 3277    u. cun 3278    i^i cin 3279    C_ wss 3280   (/)c0 3588   {csn 3774   class class class wbr 4172   omcom 4804   `'ccnv 4836   ran crn 4838    |` cres 4839   "cima 4840   Fun wfun 5407    Fn wfn 5408   -->wf 5409   -1-1->wf1 5410   -1-1-onto->wf1o 5412   ` cfv 5413    ~~ cen 7065    ~<_ cdom 7066
This theorem is referenced by:  infdiffi  7568  infcda1  8029  infpss  8053
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-1o 6683  df-er 6864  df-en 7069  df-dom 7070
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