Theorem negfi 10992

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 3086	. . . . . . . . . 10 |- ( A C_ RR -> ( a e. A -> a e. RR ) )
2		renegcl 8016	. . . . . . . . . 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 2503	. . . . . . 7 |- ( A C_ RR -> A. a e. A -u a e. RR )
6		dmmptg 5031	. . . . . . 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 2143	. . . . 5 |- ( A C_ RR -> A = dom ( a e. A |-> -u a ) )
9	8	eleq1d 2206	. . . 4 |- ( A C_ RR -> ( A e. Fin <-> dom ( a e. A |-> -u a ) e. Fin ) )
10		funmpt 5156	. . . . 5 |- Fun ( a e. A |-> -u a )
11		fundmfibi 6820	. . . . 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 7747	. . . . . 6 |- RR e. _V
15	14	ssex 4060	. . . . 5 |- ( A C_ RR -> A e. _V )
16		mptexg 5638	. . . . 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 2137	. . . . . 6 |- ( a e. A |-> -u a ) = ( a e. A |-> -u a )
19	18	negf1o 8137	. . . . 5 |- ( A C_ RR -> ( a e. A |-> -u a ) : A -1-1-onto-> { x e. RR | -u x e. A } )
20		f1of1 5359	. . . . 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 6826	. . . 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 408	. . 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 7746	. . . . . . . . . . . . . . . . 17 |- ( a e. RR -> a e. CC )
27	26	negnegd 8057	. . . . . . . . . . . . . . . 16 |- ( a e. RR -> -u -u a = a )
28	27	eqcomd 2143	. . . . . . . . . . . . . . 15 |- ( a e. RR -> a = -u -u a )
29	28	eleq1d 2206	. . . . . . . . . . . . . 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 417	. . . . . . . . 9 |- ( ( A C_ RR /\ a e. A ) -> ( -u a e. RR /\ -u -u a e. A ) )
35		eleq1 2200	. . . . . . . . . 10 |- ( n = -u a -> ( n e. RR <-> -u a e. RR ) )
36		negeq 7948	. . . . . . . . . . 11 |- ( n = -u a -> -u n = -u -u a )
37	36	eleq1d 2206	. . . . . . . . . 10 |- ( n = -u a -> ( -u n e. A <-> -u -u a e. A ) )
38	35, 37	anbi12d 464	. . . . . . . . 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 2547	. . . . . . 7 |- ( A C_ RR -> ( E. a e. A n = -u a -> ( n e. RR /\ -u n e. A ) ) )
41		simprr 521	. . . . . . . . 9 |- ( ( A C_ RR /\ ( n e. RR /\ -u n e. A ) ) -> -u n e. A )
42		negeq 7948	. . . . . . . . . . 11 |- ( a = -u n -> -u a = -u -u n )
43	42	eqeq2d 2149	. . . . . . . . . 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 7746	. . . . . . . . . . 11 |- ( n e. RR -> n e. CC )
46		negneg 8005	. . . . . . . . . . . 12 |- ( n e. CC -> -u -u n = n )
47	46	eqcomd 2143	. . . . . . . . . . 11 |- ( n e. CC -> n = -u -u n )
48	45, 47	syl 14	. . . . . . . . . 10 |- ( n e. RR -> n = -u -u n )
49	48	ad2antrl 481	. . . . . . . . 9 |- ( ( A C_ RR /\ ( n e. RR /\ -u n e. A ) ) -> n = -u -u n )
50	41, 44, 49	rspcedvd 2790	. . . . . . . 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 2255	. . . . 5 |- ( A C_ RR -> { n | E. a e. A n = -u a } = { n | ( n e. RR /\ -u n e. A ) } )
54	18	rnmpt 4782	. . . . 5 |- ran ( a e. A |-> -u a ) = { n | E. a e. A n = -u a }
55		df-rab 2423	. . . . 5 |- { n e. RR | -u n e. A } = { n | ( n e. RR /\ -u n e. A ) }
56	53, 54, 55	3eqtr4g 2195	. . . 4 |- ( A C_ RR -> ran ( a e. A |-> -u a ) = { n e. RR | -u n e. A } )
57	56	eleq1d 2206	. . 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 1331   e. wcel 1480   {cab 2123   A.wral 2414   E.wrex 2415   {crab 2418   _Vcvv 2681   C_ wss 3066   |-> cmpt 3984   dom cdm 4534   ran crn 4535   Fun wfun 5112   -1-1->wf1 5115   -1-1-onto->wf1o 5117   Fincfn 6627   CCcc 7611   RRcr 7612   -ucneg 7927

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-1o 6306  df-er 6422  df-en 6628  df-fin 6630  df-sub 7928  df-neg 7929

This theorem is referenced by: (None)