Theorem negf1o 8699

Description: Negation is an isomorphism of a subset of the real numbers to the negated elements of the subset. (Contributed by AV, 9-Aug-2020.)

Hypothesis

Ref	Expression
negf1o.1	|- F = ( x e. A |-> -u x )

Assertion

Ref	Expression
negf1o	|- ( A C_ RR -> F : A -1-1-onto-> { n e. RR | -u n e. A } )

Distinct variable group:   A, n, x

Allowed substitution hints:   F( x, n)

Proof of Theorem negf1o

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		negf1o.1	. . 3 |- F = ( x e. A |-> -u x )
2		ssel 3242	. . . . . 6 |- ( A C_ RR -> ( x e. A -> x e. RR ) )
3		renegcl 8577	. . . . . 6 |- ( x e. RR -> -u x e. RR )
4	2, 3	syl6 33	. . . . 5 |- ( A C_ RR -> ( x e. A -> -u x e. RR ) )
5	4	imp 124	. . . 4 |- ( ( A C_ RR /\ x e. A ) -> -u x e. RR )
6	2	imp 124	. . . . 5 |- ( ( A C_ RR /\ x e. A ) -> x e. RR )
7		recn 8302	. . . . . . . . 9 |- ( x e. RR -> x e. CC )
8		negneg 8566	. . . . . . . . . 10 |- ( x e. CC -> -u -u x = x )
9	8	eqcomd 2244	. . . . . . . . 9 |- ( x e. CC -> x = -u -u x )
10	7, 9	syl 14	. . . . . . . 8 |- ( x e. RR -> x = -u -u x )
11	10	eleq1d 2307	. . . . . . 7 |- ( x e. RR -> ( x e. A <-> -u -u x e. A ) )
12	11	biimpcd 159	. . . . . 6 |- ( x e. A -> ( x e. RR -> -u -u x e. A ) )
13	12	adantl 277	. . . . 5 |- ( ( A C_ RR /\ x e. A ) -> ( x e. RR -> -u -u x e. A ) )
14	6, 13	mpd 13	. . . 4 |- ( ( A C_ RR /\ x e. A ) -> -u -u x e. A )
15		negeq 8509	. . . . . 6 |- ( n = -u x -> -u n = -u -u x )
16	15	eleq1d 2307	. . . . 5 |- ( n = -u x -> ( -u n e. A <-> -u -u x e. A ) )
17	16	elrab 2982	. . . 4 |- ( -u x e. { n e. RR | -u n e. A } <-> ( -u x e. RR /\ -u -u x e. A ) )
18	5, 14, 17	sylanbrc 421	. . 3 |- ( ( A C_ RR /\ x e. A ) -> -u x e. { n e. RR | -u n e. A } )
19		negeq 8509	. . . . . . 7 |- ( n = y -> -u n = -u y )
20	19	eleq1d 2307	. . . . . 6 |- ( n = y -> ( -u n e. A <-> -u y e. A ) )
21	20	elrab 2982	. . . . 5 |- ( y e. { n e. RR | -u n e. A } <-> ( y e. RR /\ -u y e. A ) )
22		simpr 110	. . . . . 6 |- ( ( y e. RR /\ -u y e. A ) -> -u y e. A )
23	22	a1i 9	. . . . 5 |- ( A C_ RR -> ( ( y e. RR /\ -u y e. A ) -> -u y e. A ) )
24	21, 23	biimtrid 152	. . . 4 |- ( A C_ RR -> ( y e. { n e. RR | -u n e. A } -> -u y e. A ) )
25	24	imp 124	. . 3 |- ( ( A C_ RR /\ y e. { n e. RR | -u n e. A } ) -> -u y e. A )
26	2, 7	syl6com 35	. . . . . . . . . 10 |- ( x e. A -> ( A C_ RR -> x e. CC ) )
27	26	adantl 277	. . . . . . . . 9 |- ( ( ( y e. RR /\ -u y e. A ) /\ x e. A ) -> ( A C_ RR -> x e. CC ) )
28	27	imp 124	. . . . . . . 8 |- ( ( ( ( y e. RR /\ -u y e. A ) /\ x e. A ) /\ A C_ RR ) -> x e. CC )
29		recn 8302	. . . . . . . . 9 |- ( y e. RR -> y e. CC )
30	29	ad3antrrr 496	. . . . . . . 8 |- ( ( ( ( y e. RR /\ -u y e. A ) /\ x e. A ) /\ A C_ RR ) -> y e. CC )
31		negcon2 8569	. . . . . . . 8 |- ( ( x e. CC /\ y e. CC ) -> ( x = -u y <-> y = -u x ) )
32	28, 30, 31	syl2anc 415	. . . . . . 7 |- ( ( ( ( y e. RR /\ -u y e. A ) /\ x e. A ) /\ A C_ RR ) -> ( x = -u y <-> y = -u x ) )
33	32	exp31 364	. . . . . 6 |- ( ( y e. RR /\ -u y e. A ) -> ( x e. A -> ( A C_ RR -> ( x = -u y <-> y = -u x ) ) ) )
34	21, 33	sylbi 121	. . . . 5 |- ( y e. { n e. RR | -u n e. A } -> ( x e. A -> ( A C_ RR -> ( x = -u y <-> y = -u x ) ) ) )
35	34	impcom 125	. . . 4 |- ( ( x e. A /\ y e. { n e. RR | -u n e. A } ) -> ( A C_ RR -> ( x = -u y <-> y = -u x ) ) )
36	35	impcom 125	. . 3 |- ( ( A C_ RR /\ ( x e. A /\ y e. { n e. RR | -u n e. A } ) ) -> ( x = -u y <-> y = -u x ) )
37	1, 18, 25, 36	f1ocnv2d 6284	. 2 |- ( A C_ RR -> ( F : A -1-1-onto-> { n e. RR | -u n e. A } /\ `' F = ( y e. { n e. RR | -u n e. A } |-> -u y ) ) )
38	37	simpld 112	1 |- ( A C_ RR -> F : A -1-1-onto-> { n e. RR | -u n e. A } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {crab 2532   C_ wss 3220   |-> cmpt 4187   `'ccnv 4768   -1-1-onto->wf1o 5371   CCcc 8167   RRcr 8168   -ucneg 8488

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-setind 4679  ax-resscn 8261  ax-1cn 8262  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-sub 8489  df-neg 8490

This theorem is referenced by:  negfi  11972