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Theorem negfi 11238
Description: The negation of a finite set of real numbers is finite. (Contributed by AV, 9-Aug-2020.)
Assertion
Ref Expression
negfi  |-  ( ( A  C_  RR  /\  A  e.  Fin )  ->  { n  e.  RR  |  -u n  e.  A }  e.  Fin )
Distinct variable group:    A, n

Proof of Theorem negfi
Dummy variables  a  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3151 . . . . . . . . . 10  |-  ( A 
C_  RR  ->  ( a  e.  A  ->  a  e.  RR ) )
2 renegcl 8220 . . . . . . . . . 10  |-  ( a  e.  RR  ->  -u a  e.  RR )
31, 2syl6 33 . . . . . . . . 9  |-  ( A 
C_  RR  ->  ( a  e.  A  ->  -u a  e.  RR ) )
43imp 124 . . . . . . . 8  |-  ( ( A  C_  RR  /\  a  e.  A )  ->  -u a  e.  RR )
54ralrimiva 2550 . . . . . . 7  |-  ( A 
C_  RR  ->  A. a  e.  A  -u a  e.  RR )
6 dmmptg 5128 . . . . . . 7  |-  ( A. a  e.  A  -u a  e.  RR  ->  dom  ( a  e.  A  |->  -u a
)  =  A )
75, 6syl 14 . . . . . 6  |-  ( A 
C_  RR  ->  dom  (
a  e.  A  |->  -u a )  =  A )
87eqcomd 2183 . . . . 5  |-  ( A 
C_  RR  ->  A  =  dom  ( a  e.  A  |->  -u a ) )
98eleq1d 2246 . . . 4  |-  ( A 
C_  RR  ->  ( A  e.  Fin  <->  dom  ( a  e.  A  |->  -u a
)  e.  Fin )
)
10 funmpt 5256 . . . . 5  |-  Fun  (
a  e.  A  |->  -u a )
11 fundmfibi 6940 . . . . 5  |-  ( Fun  ( a  e.  A  |-> 
-u a )  -> 
( ( a  e.  A  |->  -u a )  e. 
Fin 
<->  dom  ( a  e.  A  |->  -u a )  e. 
Fin ) )
1210, 11mp1i 10 . . . 4  |-  ( A 
C_  RR  ->  ( ( a  e.  A  |->  -u a )  e.  Fin  <->  dom  ( a  e.  A  |-> 
-u a )  e. 
Fin ) )
139, 12bitr4d 191 . . 3  |-  ( A 
C_  RR  ->  ( A  e.  Fin  <->  ( a  e.  A  |->  -u a
)  e.  Fin )
)
14 reex 7947 . . . . . 6  |-  RR  e.  _V
1514ssex 4142 . . . . 5  |-  ( A 
C_  RR  ->  A  e. 
_V )
16 mptexg 5743 . . . . 5  |-  ( A  e.  _V  ->  (
a  e.  A  |->  -u a )  e.  _V )
1715, 16syl 14 . . . 4  |-  ( A 
C_  RR  ->  ( a  e.  A  |->  -u a
)  e.  _V )
18 eqid 2177 . . . . . 6  |-  ( a  e.  A  |->  -u a
)  =  ( a  e.  A  |->  -u a
)
1918negf1o 8341 . . . . 5  |-  ( A 
C_  RR  ->  ( a  e.  A  |->  -u a
) : A -1-1-onto-> { x  e.  RR  |  -u x  e.  A } )
20 f1of1 5462 . . . . 5  |-  ( ( a  e.  A  |->  -u a ) : A -1-1-onto-> {
x  e.  RR  |  -u x  e.  A }  ->  ( a  e.  A  |-> 
-u a ) : A -1-1-> { x  e.  RR  |  -u x  e.  A } )
2119, 20syl 14 . . . 4  |-  ( A 
C_  RR  ->  ( a  e.  A  |->  -u a
) : A -1-1-> {
x  e.  RR  |  -u x  e.  A }
)
22 f1vrnfibi 6946 . . . 4  |-  ( ( ( a  e.  A  |-> 
-u a )  e. 
_V  /\  ( a  e.  A  |->  -u a
) : A -1-1-> {
x  e.  RR  |  -u x  e.  A }
)  ->  ( (
a  e.  A  |->  -u a )  e.  Fin  <->  ran  ( a  e.  A  |-> 
-u a )  e. 
Fin ) )
2317, 21, 22syl2anc 411 . . 3  |-  ( A 
C_  RR  ->  ( ( a  e.  A  |->  -u a )  e.  Fin  <->  ran  ( a  e.  A  |-> 
-u a )  e. 
Fin ) )
241imp 124 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  a  e.  A )  ->  a  e.  RR )
252adantl 277 . . . . . . . . . . 11  |-  ( ( ( A  C_  RR  /\  a  e.  A )  /\  a  e.  RR )  ->  -u a  e.  RR )
26 recn 7946 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  RR  ->  a  e.  CC )
2726negnegd 8261 . . . . . . . . . . . . . . . 16  |-  ( a  e.  RR  ->  -u -u a  =  a )
2827eqcomd 2183 . . . . . . . . . . . . . . 15  |-  ( a  e.  RR  ->  a  =  -u -u a )
2928eleq1d 2246 . . . . . . . . . . . . . 14  |-  ( a  e.  RR  ->  (
a  e.  A  <->  -u -u a  e.  A ) )
3029biimpcd 159 . . . . . . . . . . . . 13  |-  ( a  e.  A  ->  (
a  e.  RR  ->  -u -u a  e.  A ) )
3130adantl 277 . . . . . . . . . . . 12  |-  ( ( A  C_  RR  /\  a  e.  A )  ->  (
a  e.  RR  ->  -u -u a  e.  A ) )
3231imp 124 . . . . . . . . . . 11  |-  ( ( ( A  C_  RR  /\  a  e.  A )  /\  a  e.  RR )  ->  -u -u a  e.  A
)
3325, 32jca 306 . . . . . . . . . 10  |-  ( ( ( A  C_  RR  /\  a  e.  A )  /\  a  e.  RR )  ->  ( -u a  e.  RR  /\  -u -u a  e.  A ) )
3424, 33mpdan 421 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  a  e.  A )  ->  ( -u a  e.  RR  /\  -u -u a  e.  A
) )
35 eleq1 2240 . . . . . . . . . 10  |-  ( n  =  -u a  ->  (
n  e.  RR  <->  -u a  e.  RR ) )
36 negeq 8152 . . . . . . . . . . 11  |-  ( n  =  -u a  ->  -u n  =  -u -u a )
3736eleq1d 2246 . . . . . . . . . 10  |-  ( n  =  -u a  ->  ( -u n  e.  A  <->  -u -u a  e.  A ) )
3835, 37anbi12d 473 . . . . . . . . 9  |-  ( n  =  -u a  ->  (
( n  e.  RR  /\  -u n  e.  A
)  <->  ( -u a  e.  RR  /\  -u -u a  e.  A ) ) )
3934, 38syl5ibrcom 157 . . . . . . . 8  |-  ( ( A  C_  RR  /\  a  e.  A )  ->  (
n  =  -u a  ->  ( n  e.  RR  /\  -u n  e.  A
) ) )
4039rexlimdva 2594 . . . . . . 7  |-  ( A 
C_  RR  ->  ( E. a  e.  A  n  =  -u a  ->  (
n  e.  RR  /\  -u n  e.  A ) ) )
41 simprr 531 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  (
n  e.  RR  /\  -u n  e.  A ) )  ->  -u n  e.  A )
42 negeq 8152 . . . . . . . . . . 11  |-  ( a  =  -u n  ->  -u a  =  -u -u n )
4342eqeq2d 2189 . . . . . . . . . 10  |-  ( a  =  -u n  ->  (
n  =  -u a  <->  n  =  -u -u n ) )
4443adantl 277 . . . . . . . . 9  |-  ( ( ( A  C_  RR  /\  ( n  e.  RR  /\  -u n  e.  A
) )  /\  a  =  -u n )  -> 
( n  =  -u a 
<->  n  =  -u -u n
) )
45 recn 7946 . . . . . . . . . . 11  |-  ( n  e.  RR  ->  n  e.  CC )
46 negneg 8209 . . . . . . . . . . . 12  |-  ( n  e.  CC  ->  -u -u n  =  n )
4746eqcomd 2183 . . . . . . . . . . 11  |-  ( n  e.  CC  ->  n  =  -u -u n )
4845, 47syl 14 . . . . . . . . . 10  |-  ( n  e.  RR  ->  n  =  -u -u n )
4948ad2antrl 490 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  (
n  e.  RR  /\  -u n  e.  A ) )  ->  n  =  -u -u n )
5041, 44, 49rspcedvd 2849 . . . . . . . 8  |-  ( ( A  C_  RR  /\  (
n  e.  RR  /\  -u n  e.  A ) )  ->  E. a  e.  A  n  =  -u a )
5150ex 115 . . . . . . 7  |-  ( A 
C_  RR  ->  ( ( n  e.  RR  /\  -u n  e.  A )  ->  E. a  e.  A  n  =  -u a ) )
5240, 51impbid 129 . . . . . 6  |-  ( A 
C_  RR  ->  ( E. a  e.  A  n  =  -u a  <->  ( n  e.  RR  /\  -u n  e.  A ) ) )
5352abbidv 2295 . . . . 5  |-  ( A 
C_  RR  ->  { n  |  E. a  e.  A  n  =  -u a }  =  { n  |  ( n  e.  RR  /\  -u n  e.  A
) } )
5418rnmpt 4877 . . . . 5  |-  ran  (
a  e.  A  |->  -u a )  =  {
n  |  E. a  e.  A  n  =  -u a }
55 df-rab 2464 . . . . 5  |-  { n  e.  RR  |  -u n  e.  A }  =  {
n  |  ( n  e.  RR  /\  -u n  e.  A ) }
5653, 54, 553eqtr4g 2235 . . . 4  |-  ( A 
C_  RR  ->  ran  (
a  e.  A  |->  -u a )  =  {
n  e.  RR  |  -u n  e.  A }
)
5756eleq1d 2246 . . 3  |-  ( A 
C_  RR  ->  ( ran  ( a  e.  A  |-> 
-u a )  e. 
Fin 
<->  { n  e.  RR  |  -u n  e.  A }  e.  Fin )
)
5813, 23, 573bitrd 214 . 2  |-  ( A 
C_  RR  ->  ( A  e.  Fin  <->  { n  e.  RR  |  -u n  e.  A }  e.  Fin ) )
5958biimpa 296 1  |-  ( ( A  C_  RR  /\  A  e.  Fin )  ->  { n  e.  RR  |  -u n  e.  A }  e.  Fin )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   {crab 2459   _Vcvv 2739    C_ wss 3131    |-> cmpt 4066   dom cdm 4628   ran crn 4629   Fun wfun 5212   -1-1->wf1 5215   -1-1-onto->wf1o 5217   Fincfn 6742   CCcc 7811   RRcr 7812   -ucneg 8131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-1o 6419  df-er 6537  df-en 6743  df-fin 6745  df-sub 8132  df-neg 8133
This theorem is referenced by: (None)
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