Theorem negfi 11169

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.

Step	Hyp	Ref	Expression
1		ssel 3136	. . . . . . . . . 10 |- ( A C_ RR -> ( a e. A -> a e. RR ) )
2		renegcl 8159	. . . . . . . . . 10 |- ( a e. RR -> -u a e. RR )
3	1, 2	syl6 33	. . . . . . . . 9 |- ( A C_ RR -> ( a e. A -> -u a e. RR ) )
4	3	imp 123	. . . . . . . 8 |- ( ( A C_ RR /\ a e. A ) -> -u a e. RR )
5	4	ralrimiva 2539	. . . . . . 7 |- ( A C_ RR -> A. a e. A -u a e. RR )
6		dmmptg 5101	. . . . . . 7 |- ( A. a e. A -u a e. RR -> dom ( a e. A |-> -u a ) = A )
7	5, 6	syl 14	. . . . . 6 |- ( A C_ RR -> dom ( a e. A |-> -u a ) = A )
8	7	eqcomd 2171	. . . . 5 |- ( A C_ RR -> A = dom ( a e. A |-> -u a ) )
9	8	eleq1d 2235	. . . 4 |- ( A C_ RR -> ( A e. Fin <-> dom ( a e. A |-> -u a ) e. Fin ) )
10		funmpt 5226	. . . . 5 |- Fun ( a e. A |-> -u a )
11		fundmfibi 6904	. . . . 5 |- ( Fun ( a e. A |-> -u a ) -> ( ( a e. A |-> -u a ) e. Fin <-> dom ( a e. A |-> -u a ) e. Fin ) )
12	10, 11	mp1i 10	. . . 4 |- ( A C_ RR -> ( ( a e. A |-> -u a ) e. Fin <-> dom ( a e. A |-> -u a ) e. Fin ) )
13	9, 12	bitr4d 190	. . 3 |- ( A C_ RR -> ( A e. Fin <-> ( a e. A |-> -u a ) e. Fin ) )
14		reex 7887	. . . . . 6 |- RR e. _V
15	14	ssex 4119	. . . . 5 |- ( A C_ RR -> A e. _V )
16		mptexg 5710	. . . . 5 |- ( A e. _V -> ( a e. A |-> -u a ) e. _V )
17	15, 16	syl 14	. . . 4 |- ( A C_ RR -> ( a e. A |-> -u a ) e. _V )
18		eqid 2165	. . . . . 6 |- ( a e. A |-> -u a ) = ( a e. A |-> -u a )
19	18	negf1o 8280	. . . . 5 |- ( A C_ RR -> ( a e. A |-> -u a ) : A -1-1-onto-> { x e. RR | -u x e. A } )
20		f1of1 5431	. . . . 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 } )
21	19, 20	syl 14	. . . 4 |- ( A C_ RR -> ( a e. A |-> -u a ) : A -1-1-> { x e. RR | -u x e. A } )
22		f1vrnfibi 6910	. . . 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 ) )
23	17, 21, 22	syl2anc 409	. . 3 |- ( A C_ RR -> ( ( a e. A |-> -u a ) e. Fin <-> ran ( a e. A |-> -u a ) e. Fin ) )
24	1	imp 123	. . . . . . . . . 10 |- ( ( A C_ RR /\ a e. A ) -> a e. RR )
25	2	adantl 275	. . . . . . . . . . 11 |- ( ( ( A C_ RR /\ a e. A ) /\ a e. RR ) -> -u a e. RR )
26		recn 7886	. . . . . . . . . . . . . . . . 17 |- ( a e. RR -> a e. CC )
27	26	negnegd 8200	. . . . . . . . . . . . . . . 16 |- ( a e. RR -> -u -u a = a )
28	27	eqcomd 2171	. . . . . . . . . . . . . . 15 |- ( a e. RR -> a = -u -u a )
29	28	eleq1d 2235	. . . . . . . . . . . . . 14 |- ( a e. RR -> ( a e. A <-> -u -u a e. A ) )
30	29	biimpcd 158	. . . . . . . . . . . . 13 |- ( a e. A -> ( a e. RR -> -u -u a e. A ) )
31	30	adantl 275	. . . . . . . . . . . 12 |- ( ( A C_ RR /\ a e. A ) -> ( a e. RR -> -u -u a e. A ) )
32	31	imp 123	. . . . . . . . . . 11 |- ( ( ( A C_ RR /\ a e. A ) /\ a e. RR ) -> -u -u a e. A )
33	25, 32	jca 304	. . . . . . . . . 10 |- ( ( ( A C_ RR /\ a e. A ) /\ a e. RR ) -> ( -u a e. RR /\ -u -u a e. A ) )
34	24, 33	mpdan 418	. . . . . . . . 9 |- ( ( A C_ RR /\ a e. A ) -> ( -u a e. RR /\ -u -u a e. A ) )
35		eleq1 2229	. . . . . . . . . 10 |- ( n = -u a -> ( n e. RR <-> -u a e. RR ) )
36		negeq 8091	. . . . . . . . . . 11 |- ( n = -u a -> -u n = -u -u a )
37	36	eleq1d 2235	. . . . . . . . . 10 |- ( n = -u a -> ( -u n e. A <-> -u -u a e. A ) )
38	35, 37	anbi12d 465	. . . . . . . . 9 |- ( n = -u a -> ( ( n e. RR /\ -u n e. A ) <-> ( -u a e. RR /\ -u -u a e. A ) ) )
39	34, 38	syl5ibrcom 156	. . . . . . . 8 |- ( ( A C_ RR /\ a e. A ) -> ( n = -u a -> ( n e. RR /\ -u n e. A ) ) )
40	39	rexlimdva 2583	. . . . . . 7 |- ( A C_ RR -> ( E. a e. A n = -u a -> ( n e. RR /\ -u n e. A ) ) )
41		simprr 522	. . . . . . . . 9 |- ( ( A C_ RR /\ ( n e. RR /\ -u n e. A ) ) -> -u n e. A )
42		negeq 8091	. . . . . . . . . . 11 |- ( a = -u n -> -u a = -u -u n )
43	42	eqeq2d 2177	. . . . . . . . . 10 |- ( a = -u n -> ( n = -u a <-> n = -u -u n ) )
44	43	adantl 275	. . . . . . . . 9 |- ( ( ( A C_ RR /\ ( n e. RR /\ -u n e. A ) ) /\ a = -u n ) -> ( n = -u a <-> n = -u -u n ) )
45		recn 7886	. . . . . . . . . . 11 |- ( n e. RR -> n e. CC )
46		negneg 8148	. . . . . . . . . . . 12 |- ( n e. CC -> -u -u n = n )
47	46	eqcomd 2171	. . . . . . . . . . 11 |- ( n e. CC -> n = -u -u n )
48	45, 47	syl 14	. . . . . . . . . 10 |- ( n e. RR -> n = -u -u n )
49	48	ad2antrl 482	. . . . . . . . 9 |- ( ( A C_ RR /\ ( n e. RR /\ -u n e. A ) ) -> n = -u -u n )
50	41, 44, 49	rspcedvd 2836	. . . . . . . 8 |- ( ( A C_ RR /\ ( n e. RR /\ -u n e. A ) ) -> E. a e. A n = -u a )
51	50	ex 114	. . . . . . 7 |- ( A C_ RR -> ( ( n e. RR /\ -u n e. A ) -> E. a e. A n = -u a ) )
52	40, 51	impbid 128	. . . . . 6 |- ( A C_ RR -> ( E. a e. A n = -u a <-> ( n e. RR /\ -u n e. A ) ) )
53	52	abbidv 2284	. . . . 5 |- ( A C_ RR -> { n | E. a e. A n = -u a } = { n | ( n e. RR /\ -u n e. A ) } )
54	18	rnmpt 4852	. . . . 5 |- ran ( a e. A |-> -u a ) = { n | E. a e. A n = -u a }
55		df-rab 2453	. . . . 5 |- { n e. RR | -u n e. A } = { n | ( n e. RR /\ -u n e. A ) }
56	53, 54, 55	3eqtr4g 2224	. . . 4 |- ( A C_ RR -> ran ( a e. A |-> -u a ) = { n e. RR | -u n e. A } )
57	56	eleq1d 2235	. . 3 |- ( A C_ RR -> ( ran ( a e. A |-> -u a ) e. Fin <-> { n e. RR | -u n e. A } e. Fin ) )
58	13, 23, 57	3bitrd 213	. 2 |- ( A C_ RR -> ( A e. Fin <-> { n e. RR | -u n e. A } e. Fin ) )
59	58	biimpa 294	1 |- ( ( A C_ RR /\ A e. Fin ) -> { n e. RR | -u n e. A } e. Fin )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   {cab 2151   A.wral 2444   E.wrex 2445   {crab 2448   _Vcvv 2726   C_ wss 3116   |-> cmpt 4043   dom cdm 4604   ran crn 4605   Fun wfun 5182   -1-1->wf1 5185   -1-1-onto->wf1o 5187   Fincfn 6706   CCcc 7751   RRcr 7752   -ucneg 8070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-1o 6384  df-er 6501  df-en 6707  df-fin 6709  df-sub 8071  df-neg 8072

This theorem is referenced by: (None)