Theorem negf1o 8329

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 ∈ 𝐴 ↦ -𝑥)

Assertion

Ref	Expression
negf1o	⊢ (𝐴 ⊆ ℝ → 𝐹:𝐴–1-1-onto→{𝑛 ∈ ℝ ∣ -𝑛 ∈ 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 ∈ 𝐴 ↦ -𝑥)
2		ssel 3149	. . . . . 6 ⊢ (𝐴 ⊆ ℝ → (𝑥 ∈ 𝐴 → 𝑥 ∈ ℝ))
3		renegcl 8208	. . . . . 6 ⊢ (𝑥 ∈ ℝ → -𝑥 ∈ ℝ)
4	2, 3	syl6 33	. . . . 5 ⊢ (𝐴 ⊆ ℝ → (𝑥 ∈ 𝐴 → -𝑥 ∈ ℝ))
5	4	imp 124	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝐴) → -𝑥 ∈ ℝ)
6	2	imp 124	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
7		recn 7935	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
8		negneg 8197	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → --𝑥 = 𝑥)
9	8	eqcomd 2183	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → 𝑥 = --𝑥)
10	7, 9	syl 14	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → 𝑥 = --𝑥)
11	10	eleq1d 2246	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → (𝑥 ∈ 𝐴 ↔ --𝑥 ∈ 𝐴))
12	11	biimpcd 159	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ ℝ → --𝑥 ∈ 𝐴))
13	12	adantl 277	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ ℝ → --𝑥 ∈ 𝐴))
14	6, 13	mpd 13	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝐴) → --𝑥 ∈ 𝐴)
15		negeq 8140	. . . . . 6 ⊢ (𝑛 = -𝑥 → -𝑛 = --𝑥)
16	15	eleq1d 2246	. . . . 5 ⊢ (𝑛 = -𝑥 → (-𝑛 ∈ 𝐴 ↔ --𝑥 ∈ 𝐴))
17	16	elrab 2893	. . . 4 ⊢ (-𝑥 ∈ {𝑛 ∈ ℝ ∣ -𝑛 ∈ 𝐴} ↔ (-𝑥 ∈ ℝ ∧ --𝑥 ∈ 𝐴))
18	5, 14, 17	sylanbrc 417	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝐴) → -𝑥 ∈ {𝑛 ∈ ℝ ∣ -𝑛 ∈ 𝐴})
19		negeq 8140	. . . . . . 7 ⊢ (𝑛 = 𝑦 → -𝑛 = -𝑦)
20	19	eleq1d 2246	. . . . . 6 ⊢ (𝑛 = 𝑦 → (-𝑛 ∈ 𝐴 ↔ -𝑦 ∈ 𝐴))
21	20	elrab 2893	. . . . 5 ⊢ (𝑦 ∈ {𝑛 ∈ ℝ ∣ -𝑛 ∈ 𝐴} ↔ (𝑦 ∈ ℝ ∧ -𝑦 ∈ 𝐴))
22		simpr 110	. . . . . 6 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ 𝐴) → -𝑦 ∈ 𝐴)
23	22	a1i 9	. . . . 5 ⊢ (𝐴 ⊆ ℝ → ((𝑦 ∈ ℝ ∧ -𝑦 ∈ 𝐴) → -𝑦 ∈ 𝐴))
24	21, 23	biimtrid 152	. . . 4 ⊢ (𝐴 ⊆ ℝ → (𝑦 ∈ {𝑛 ∈ ℝ ∣ -𝑛 ∈ 𝐴} → -𝑦 ∈ 𝐴))
25	24	imp 124	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ {𝑛 ∈ ℝ ∣ -𝑛 ∈ 𝐴}) → -𝑦 ∈ 𝐴)
26	2, 7	syl6com 35	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (𝐴 ⊆ ℝ → 𝑥 ∈ ℂ))
27	26	adantl 277	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ ∧ -𝑦 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝐴 ⊆ ℝ → 𝑥 ∈ ℂ))
28	27	imp 124	. . . . . . . 8 ⊢ ((((𝑦 ∈ ℝ ∧ -𝑦 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝐴 ⊆ ℝ) → 𝑥 ∈ ℂ)
29		recn 7935	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
30	29	ad3antrrr 492	. . . . . . . 8 ⊢ ((((𝑦 ∈ ℝ ∧ -𝑦 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝐴 ⊆ ℝ) → 𝑦 ∈ ℂ)
31		negcon2 8200	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥))
32	28, 30, 31	syl2anc 411	. . . . . . 7 ⊢ ((((𝑦 ∈ ℝ ∧ -𝑦 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝐴 ⊆ ℝ) → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥))
33	32	exp31 364	. . . . . 6 ⊢ ((𝑦 ∈ ℝ ∧ -𝑦 ∈ 𝐴) → (𝑥 ∈ 𝐴 → (𝐴 ⊆ ℝ → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥))))
34	21, 33	sylbi 121	. . . . 5 ⊢ (𝑦 ∈ {𝑛 ∈ ℝ ∣ -𝑛 ∈ 𝐴} → (𝑥 ∈ 𝐴 → (𝐴 ⊆ ℝ → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥))))
35	34	impcom 125	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑛 ∈ ℝ ∣ -𝑛 ∈ 𝐴}) → (𝐴 ⊆ ℝ → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥)))
36	35	impcom 125	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑛 ∈ ℝ ∣ -𝑛 ∈ 𝐴})) → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥))
37	1, 18, 25, 36	f1ocnv2d 6069	. 2 ⊢ (𝐴 ⊆ ℝ → (𝐹:𝐴–1-1-onto→{𝑛 ∈ ℝ ∣ -𝑛 ∈ 𝐴} ∧ ◡𝐹 = (𝑦 ∈ {𝑛 ∈ ℝ ∣ -𝑛 ∈ 𝐴} ↦ -𝑦)))
38	37	simpld 112	1 ⊢ (𝐴 ⊆ ℝ → 𝐹:𝐴–1-1-onto→{𝑛 ∈ ℝ ∣ -𝑛 ∈ 𝐴})

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  {crab 2459   ⊆ wss 3129   ↦ cmpt 4061  ^◡ccnv 4622  –1-1-onto→wf1o 5211  ℂcc 7800  ℝcr 7801  -cneg 8119

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-setind 4533  ax-resscn 7894  ax-1cn 7895  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-sub 8120  df-neg 8121

This theorem is referenced by:  negfi  11220