Theorem negfi 11374

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 3174	. . . . . . . . . 10 |- ( A C_ RR -> ( a e. A -> a e. RR ) )
2		renegcl 8282	. . . . . . . . . 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 124	. . . . . . . 8 |- ( ( A C_ RR /\ a e. A ) -> -u a e. RR )
5	4	ralrimiva 2567	. . . . . . 7 |- ( A C_ RR -> A. a e. A -u a e. RR )
6		dmmptg 5164	. . . . . . 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 2199	. . . . 5 |- ( A C_ RR -> A = dom ( a e. A |-> -u a ) )
9	8	eleq1d 2262	. . . 4 |- ( A C_ RR -> ( A e. Fin <-> dom ( a e. A |-> -u a ) e. Fin ) )
10		funmpt 5293	. . . . 5 |- Fun ( a e. A |-> -u a )
11		fundmfibi 6999	. . . . 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 191	. . 3 |- ( A C_ RR -> ( A e. Fin <-> ( a e. A |-> -u a ) e. Fin ) )
14		reex 8008	. . . . . 6 |- RR e. _V
15	14	ssex 4167	. . . . 5 |- ( A C_ RR -> A e. _V )
16		mptexg 5784	. . . . 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 2193	. . . . . 6 |- ( a e. A |-> -u a ) = ( a e. A |-> -u a )
19	18	negf1o 8403	. . . . 5 |- ( A C_ RR -> ( a e. A |-> -u a ) : A -1-1-onto-> { x e. RR | -u x e. A } )
20		f1of1 5500	. . . . 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 7006	. . . 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 411	. . 3 |- ( A C_ RR -> ( ( a e. A |-> -u a ) e. Fin <-> ran ( a e. A |-> -u a ) e. Fin ) )
24	1	imp 124	. . . . . . . . . 10 |- ( ( A C_ RR /\ a e. A ) -> a e. RR )
25	2	adantl 277	. . . . . . . . . . 11 |- ( ( ( A C_ RR /\ a e. A ) /\ a e. RR ) -> -u a e. RR )
26		recn 8007	. . . . . . . . . . . . . . . . 17 |- ( a e. RR -> a e. CC )
27	26	negnegd 8323	. . . . . . . . . . . . . . . 16 |- ( a e. RR -> -u -u a = a )
28	27	eqcomd 2199	. . . . . . . . . . . . . . 15 |- ( a e. RR -> a = -u -u a )
29	28	eleq1d 2262	. . . . . . . . . . . . . 14 |- ( a e. RR -> ( a e. A <-> -u -u a e. A ) )
30	29	biimpcd 159	. . . . . . . . . . . . 13 |- ( a e. A -> ( a e. RR -> -u -u a e. A ) )
31	30	adantl 277	. . . . . . . . . . . 12 |- ( ( A C_ RR /\ a e. A ) -> ( a e. RR -> -u -u a e. A ) )
32	31	imp 124	. . . . . . . . . . 11 |- ( ( ( A C_ RR /\ a e. A ) /\ a e. RR ) -> -u -u a e. A )
33	25, 32	jca 306	. . . . . . . . . 10 |- ( ( ( A C_ RR /\ a e. A ) /\ a e. RR ) -> ( -u a e. RR /\ -u -u a e. A ) )
34	24, 33	mpdan 421	. . . . . . . . 9 |- ( ( A C_ RR /\ a e. A ) -> ( -u a e. RR /\ -u -u a e. A ) )
35		eleq1 2256	. . . . . . . . . 10 |- ( n = -u a -> ( n e. RR <-> -u a e. RR ) )
36		negeq 8214	. . . . . . . . . . 11 |- ( n = -u a -> -u n = -u -u a )
37	36	eleq1d 2262	. . . . . . . . . 10 |- ( n = -u a -> ( -u n e. A <-> -u -u a e. A ) )
38	35, 37	anbi12d 473	. . . . . . . . 9 |- ( n = -u a -> ( ( n e. RR /\ -u n e. A ) <-> ( -u a e. RR /\ -u -u a e. A ) ) )
39	34, 38	syl5ibrcom 157	. . . . . . . 8 |- ( ( A C_ RR /\ a e. A ) -> ( n = -u a -> ( n e. RR /\ -u n e. A ) ) )
40	39	rexlimdva 2611	. . . . . . 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 8214	. . . . . . . . . . 11 |- ( a = -u n -> -u a = -u -u n )
43	42	eqeq2d 2205	. . . . . . . . . 10 |- ( a = -u n -> ( n = -u a <-> n = -u -u n ) )
44	43	adantl 277	. . . . . . . . 9 |- ( ( ( A C_ RR /\ ( n e. RR /\ -u n e. A ) ) /\ a = -u n ) -> ( n = -u a <-> n = -u -u n ) )
45		recn 8007	. . . . . . . . . . 11 |- ( n e. RR -> n e. CC )
46		negneg 8271	. . . . . . . . . . . 12 |- ( n e. CC -> -u -u n = n )
47	46	eqcomd 2199	. . . . . . . . . . 11 |- ( n e. CC -> n = -u -u n )
48	45, 47	syl 14	. . . . . . . . . 10 |- ( n e. RR -> n = -u -u n )
49	48	ad2antrl 490	. . . . . . . . 9 |- ( ( A C_ RR /\ ( n e. RR /\ -u n e. A ) ) -> n = -u -u n )
50	41, 44, 49	rspcedvd 2871	. . . . . . . 8 |- ( ( A C_ RR /\ ( n e. RR /\ -u n e. A ) ) -> E. a e. A n = -u a )
51	50	ex 115	. . . . . . 7 |- ( A C_ RR -> ( ( n e. RR /\ -u n e. A ) -> E. a e. A n = -u a ) )
52	40, 51	impbid 129	. . . . . 6 |- ( A C_ RR -> ( E. a e. A n = -u a <-> ( n e. RR /\ -u n e. A ) ) )
53	52	abbidv 2311	. . . . 5 |- ( A C_ RR -> { n | E. a e. A n = -u a } = { n | ( n e. RR /\ -u n e. A ) } )
54	18	rnmpt 4911	. . . . 5 |- ran ( a e. A |-> -u a ) = { n | E. a e. A n = -u a }
55		df-rab 2481	. . . . 5 |- { n e. RR | -u n e. A } = { n | ( n e. RR /\ -u n e. A ) }
56	53, 54, 55	3eqtr4g 2251	. . . 4 |- ( A C_ RR -> ran ( a e. A |-> -u a ) = { n e. RR | -u n e. A } )
57	56	eleq1d 2262	. . 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 214	. 2 |- ( A C_ RR -> ( A e. Fin <-> { n e. RR | -u n e. A } e. Fin ) )
59	58	biimpa 296	1 |- ( ( A C_ RR /\ A e. Fin ) -> { n e. RR | -u n e. A } e. Fin )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   {cab 2179   A.wral 2472   E.wrex 2473   {crab 2476   _Vcvv 2760   C_ wss 3154   |-> cmpt 4091   dom cdm 4660   ran crn 4661   Fun wfun 5249   -1-1->wf1 5252   -1-1-onto->wf1o 5254   Fincfn 6796   CCcc 7872   RRcr 7873   -ucneg 8193

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-1o 6471  df-er 6589  df-en 6797  df-fin 6799  df-sub 8194  df-neg 8195

This theorem is referenced by: (None)