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Theorem infdifsn 7545
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 7056 . . . 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 7052 . . . . . . 7  |-  Rel  ~<_
43brrelex2i 4860 . . . . . 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 5580 . . . . . . 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 4805 . . . . . 6  |-  (/)  e.  om
10 ffvelrn 5808 . . . . . 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 7127 . . . . 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 2903 . . . . . . . . . 10  |-  f  e. 
_V
15 f1f1orn 5626 . . . . . . . . . . 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 7060 . . . . . . . . . 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 7095 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ran  f  ~~  om )
203brrelexi 4859 . . . . . . . . . . 11  |-  ( om  ~<_  A  ->  om  e.  _V )
2120ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  om  e.  _V )
22 limom 4801 . . . . . . . . . . 11  |-  Lim  om
2322limenpsi 7219 . . . . . . . . . 10  |-  ( om  e.  _V  ->  om  ~~  ( om  \  { (/) } ) )
2421, 23syl 16 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  om  ~~  ( om  \  { (/) } ) )
2514resex 5127 . . . . . . . . . . 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 3418 . . . . . . . . . . . 12  |-  ( om 
\  { (/) } ) 
C_  om
28 f1ores 5630 . . . . . . . . . . . 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 7060 . . . . . . . . . . 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 5625 . . . . . . . . . . . . 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 5469 . . . . . . . . . . . 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 5581 . . . . . . . . . . . . . 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 5504 . . . . . . . . . . . . 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 5726 . . . . . . . . . . . . . 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 2393 . . . . . . . . . . . 12  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( f " { (/) } )  =  { ( f `  (/) ) } )
4339, 42difeq12d 3410 . . . . . . . . . . 11  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( (
f " om )  \  ( f " { (/) } ) )  =  ( ran  f  \  { ( f `  (/) ) } ) )
4435, 43eqtrd 2420 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( f " ( om  \  { (/)
} ) )  =  ( ran  f  \  { ( f `  (/) ) } ) )
4531, 44breqtrd 4178 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( om  \  { (/) } )  ~~  ( ran  f  \  {
( f `  (/) ) } ) )
46 entr 7096 . . . . . . . . 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 7096 . . . . . . . 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 4293 . . . . . . . 8  |-  ( A  e.  _V  ->  ( A  \  ran  f )  e.  _V )
51 enrefg 7076 . . . . . . . 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 3644 . . . . . . . 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 3418 . . . . . . . . . 10  |-  ( ran  f  \  { ( f `  (/) ) } )  C_  ran  f
56 ssrin 3510 . . . . . . . . . 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 3603 . . . . . . . . 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 7126 . . . . . . 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 5538 . . . . . . . 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 3652 . . . . . . 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 3435 . . . . . . 7  |-  ( ( ran  f  \  {
( f `  (/) ) } )  u.  ( A 
\  ran  f )
)  =  ( ( A  \  ran  f
)  u.  ( ran  f  \  { ( f `  (/) ) } ) )
68 eldifn 3414 . . . . . . . . . . 11  |-  ( ( f `  (/) )  e.  ( A  \  ran  f )  ->  -.  ( f `  (/) )  e. 
ran  f )
69 fnfvelrn 5807 . . . . . . . . . . . 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 3812 . . . . . . . . . 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 3628 . . . . . . . . 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 3435 . . . . . . . . . 10  |-  ( ( A  \  ran  f
)  u.  ran  f
)  =  ( ran  f  u.  ( A 
\  ran  f )
)
7776, 66syl5eq 2432 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( ( A  \  ran  f )  u.  ran  f )  =  A )
7877difeq1d 3408 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( (
( A  \  ran  f )  u.  ran  f )  \  {
( f `  (/) ) } )  =  ( A 
\  { ( f `
 (/) ) } ) )
7975, 78eqtrd 2420 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( ( A  \  ran  f )  u.  ( ran  f  \  { ( f `  (/) ) } ) )  =  ( A  \  { ( f `  (/) ) } ) )
8067, 79syl5eq 2432 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( ( ran  f  \  { ( f `  (/) ) } )  u.  ( A 
\  ran  f )
)  =  ( A 
\  { ( f `
 (/) ) } ) )
8162, 66, 803brtr3d 4183 . . . . 5  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  A  ~~  ( A  \  { ( f `  (/) ) } ) )
8281ensymd 7095 . . . 4  |-  ( ( ( om  ~<_  A  /\  B  e.  A )  /\  f : om -1-1-> A
)  ->  ( A  \  { ( f `  (/) ) } )  ~~  A )
83 entr 7096 . . . 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 3877 . . . 4  |-  ( -.  B  e.  A  -> 
( A  \  { B } )  =  A )
8786adantl 453 . . 3  |-  ( ( om  ~<_  A  /\  -.  B  e.  A )  ->  ( A  \  { B } )  =  A )
88 enrefg 7076 . . . . 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 4174 . 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 1717   _Vcvv 2900    \ cdif 3261    u. cun 3262    i^i cin 3263    C_ wss 3264   (/)c0 3572   {csn 3758   class class class wbr 4154   omcom 4786   `'ccnv 4818   ran crn 4820    |` cres 4821   "cima 4822   Fun wfun 5389    Fn wfn 5390   -->wf 5391   -1-1->wf1 5392   -1-1-onto->wf1o 5394   ` cfv 5395    ~~ cen 7043    ~<_ cdom 7044
This theorem is referenced by:  infdiffi  7546  infcda1  8007  infpss  8031
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-1o 6661  df-er 6842  df-en 7047  df-dom 7048
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