Theorem negfi 11410

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 3178	. . . . . . . . . 10 |- ( A C_ RR -> ( a e. A -> a e. RR ) )
2		renegcl 8304	. . . . . . . . . 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 2570	. . . . . . 7 |- ( A C_ RR -> A. a e. A -u a e. RR )
6		dmmptg 5168	. . . . . . 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 2202	. . . . 5 |- ( A C_ RR -> A = dom ( a e. A |-> -u a ) )
9	8	eleq1d 2265	. . . 4 |- ( A C_ RR -> ( A e. Fin <-> dom ( a e. A |-> -u a ) e. Fin ) )
10		funmpt 5297	. . . . 5 |- Fun ( a e. A |-> -u a )
11		fundmfibi 7013	. . . . 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 8030	. . . . . 6 |- RR e. _V
15	14	ssex 4171	. . . . 5 |- ( A C_ RR -> A e. _V )
16		mptexg 5790	. . . . 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 2196	. . . . . 6 |- ( a e. A |-> -u a ) = ( a e. A |-> -u a )
19	18	negf1o 8425	. . . . 5 |- ( A C_ RR -> ( a e. A |-> -u a ) : A -1-1-onto-> { x e. RR | -u x e. A } )
20		f1of1 5506	. . . . 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 7020	. . . 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 8029	. . . . . . . . . . . . . . . . 17 |- ( a e. RR -> a e. CC )
27	26	negnegd 8345	. . . . . . . . . . . . . . . 16 |- ( a e. RR -> -u -u a = a )
28	27	eqcomd 2202	. . . . . . . . . . . . . . 15 |- ( a e. RR -> a = -u -u a )
29	28	eleq1d 2265	. . . . . . . . . . . . . 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 2259	. . . . . . . . . 10 |- ( n = -u a -> ( n e. RR <-> -u a e. RR ) )
36		negeq 8236	. . . . . . . . . . 11 |- ( n = -u a -> -u n = -u -u a )
37	36	eleq1d 2265	. . . . . . . . . 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 2614	. . . . . . 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 8236	. . . . . . . . . . 11 |- ( a = -u n -> -u a = -u -u n )
43	42	eqeq2d 2208	. . . . . . . . . 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 8029	. . . . . . . . . . 11 |- ( n e. RR -> n e. CC )
46		negneg 8293	. . . . . . . . . . . 12 |- ( n e. CC -> -u -u n = n )
47	46	eqcomd 2202	. . . . . . . . . . 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 2874	. . . . . . . 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 2314	. . . . 5 |- ( A C_ RR -> { n | E. a e. A n = -u a } = { n | ( n e. RR /\ -u n e. A ) } )
54	18	rnmpt 4915	. . . . 5 |- ran ( a e. A |-> -u a ) = { n | E. a e. A n = -u a }
55		df-rab 2484	. . . . 5 |- { n e. RR | -u n e. A } = { n | ( n e. RR /\ -u n e. A ) }
56	53, 54, 55	3eqtr4g 2254	. . . 4 |- ( A C_ RR -> ran ( a e. A |-> -u a ) = { n e. RR | -u n e. A } )
57	56	eleq1d 2265	. . 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 2167   {cab 2182   A.wral 2475   E.wrex 2476   {crab 2479   _Vcvv 2763   C_ wss 3157   |-> cmpt 4095   dom cdm 4664   ran crn 4665   Fun wfun 5253   -1-1->wf1 5256   -1-1-onto->wf1o 5258   Fincfn 6808   CCcc 7894   RRcr 7895   -ucneg 8215

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-1o 6483  df-er 6601  df-en 6809  df-fin 6811  df-sub 8216  df-neg 8217

This theorem is referenced by: (None)