Theorem negf1o 8453

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 3186	. . . . . 6 ⊢ (𝐴 ⊆ ℝ → (𝑥 ∈ 𝐴 → 𝑥 ∈ ℝ))
3		renegcl 8332	. . . . . 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 8057	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
8		negneg 8321	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → --𝑥 = 𝑥)
9	8	eqcomd 2210	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → 𝑥 = --𝑥)
10	7, 9	syl 14	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → 𝑥 = --𝑥)
11	10	eleq1d 2273	. . . . . . 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 8264	. . . . . 6 ⊢ (𝑛 = -𝑥 → -𝑛 = --𝑥)
16	15	eleq1d 2273	. . . . 5 ⊢ (𝑛 = -𝑥 → (-𝑛 ∈ 𝐴 ↔ --𝑥 ∈ 𝐴))
17	16	elrab 2928	. . . 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 8264	. . . . . . 7 ⊢ (𝑛 = 𝑦 → -𝑛 = -𝑦)
20	19	eleq1d 2273	. . . . . 6 ⊢ (𝑛 = 𝑦 → (-𝑛 ∈ 𝐴 ↔ -𝑦 ∈ 𝐴))
21	20	elrab 2928	. . . . 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 8057	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
30	29	ad3antrrr 492	. . . . . . . 8 ⊢ ((((𝑦 ∈ ℝ ∧ -𝑦 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝐴 ⊆ ℝ) → 𝑦 ∈ ℂ)
31		negcon2 8324	. . . . . . . 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 6149	. 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 1372   ∈ wcel 2175  {crab 2487   ⊆ wss 3165   ↦ cmpt 4104  ^◡ccnv 4673  –1-1-onto→wf1o 5269  ℂcc 7922  ℝcr 7923  -cneg 8243

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-setind 4584  ax-resscn 8016  ax-1cn 8017  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-distr 8028  ax-i2m1 8029  ax-0id 8032  ax-rnegex 8033  ax-cnre 8035

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-sub 8244  df-neg 8245

This theorem is referenced by:  negfi  11481