Theorem negfi 11938

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 3236	. . . . . . . . . 10 |- ( A C_ RR -> ( a e. A -> a e. RR ) )
2		renegcl 8550	. . . . . . . . . 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 2617	. . . . . . 7 |- ( A C_ RR -> A. a e. A -u a e. RR )
6		dmmptg 5265	. . . . . . 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 2240	. . . . 5 |- ( A C_ RR -> A = dom ( a e. A |-> -u a ) )
9	8	eleq1d 2303	. . . 4 |- ( A C_ RR -> ( A e. Fin <-> dom ( a e. A |-> -u a ) e. Fin ) )
10		funmpt 5395	. . . . 5 |- Fun ( a e. A |-> -u a )
11		fundmfibi 7218	. . . . 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 8277	. . . . . 6 |- RR e. _V
15	14	ssex 4252	. . . . 5 |- ( A C_ RR -> A e. _V )
16		mptexg 5916	. . . . 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 2234	. . . . . 6 |- ( a e. A |-> -u a ) = ( a e. A |-> -u a )
19	18	negf1o 8672	. . . . 5 |- ( A C_ RR -> ( a e. A |-> -u a ) : A -1-1-onto-> { x e. RR | -u x e. A } )
20		f1of1 5618	. . . . 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 7225	. . . 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 8276	. . . . . . . . . . . . . . . . 17 |- ( a e. RR -> a e. CC )
27	26	negnegd 8591	. . . . . . . . . . . . . . . 16 |- ( a e. RR -> -u -u a = a )
28	27	eqcomd 2240	. . . . . . . . . . . . . . 15 |- ( a e. RR -> a = -u -u a )
29	28	eleq1d 2303	. . . . . . . . . . . . . 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 2297	. . . . . . . . . 10 |- ( n = -u a -> ( n e. RR <-> -u a e. RR ) )
36		negeq 8482	. . . . . . . . . . 11 |- ( n = -u a -> -u n = -u -u a )
37	36	eleq1d 2303	. . . . . . . . . 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 2662	. . . . . . 7 |- ( A C_ RR -> ( E. a e. A n = -u a -> ( n e. RR /\ -u n e. A ) ) )
41		simprr 533	. . . . . . . . 9 |- ( ( A C_ RR /\ ( n e. RR /\ -u n e. A ) ) -> -u n e. A )
42		negeq 8482	. . . . . . . . . . 11 |- ( a = -u n -> -u a = -u -u n )
43	42	eqeq2d 2246	. . . . . . . . . 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 8276	. . . . . . . . . . 11 |- ( n e. RR -> n e. CC )
46		negneg 8539	. . . . . . . . . . . 12 |- ( n e. CC -> -u -u n = n )
47	46	eqcomd 2240	. . . . . . . . . . 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 2929	. . . . . . . 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 2354	. . . . 5 |- ( A C_ RR -> { n | E. a e. A n = -u a } = { n | ( n e. RR /\ -u n e. A ) } )
54	18	rnmpt 5010	. . . . 5 |- ran ( a e. A |-> -u a ) = { n | E. a e. A n = -u a }
55		df-rab 2531	. . . . 5 |- { n e. RR | -u n e. A } = { n | ( n e. RR /\ -u n e. A ) }
56	53, 54, 55	3eqtr4g 2292	. . . 4 |- ( A C_ RR -> ran ( a e. A |-> -u a ) = { n e. RR | -u n e. A } )
57	56	eleq1d 2303	. . 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 1398   e. wcel 2205   {cab 2220   A.wral 2522   E.wrex 2523   {crab 2526   _Vcvv 2815   C_ wss 3214   |-> cmpt 4176   dom cdm 4754   ran crn 4755   Fun wfun 5351   -1-1->wf1 5354   -1-1-onto->wf1o 5356   Fincfn 6988   CCcc 8141   RRcr 8142   -ucneg 8461

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-1o 6660  df-er 6780  df-en 6989  df-fin 6991  df-sub 8462  df-neg 8463

This theorem is referenced by: (None)