Theorem icoreresf 37353

Description: Closed-below, open-above intervals of reals map to subsets of reals. (Contributed by ML, 25-Jul-2020.)

Assertion

Ref	Expression
icoreresf	⊢ ([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ

Proof of Theorem icoreresf

Dummy variables 𝑥 𝑦 𝑧 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexpssxrxp 11306	. . 3 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
2		df-ico 13393	. . . . 5 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
3	2	ixxf 13397	. . . 4 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
4		ffn 6736	. . . 4 ⊢ ([,):(ℝ* × ℝ*)⟶𝒫 ℝ* → [,) Fn (ℝ* × ℝ*))
5		fnssresb 6690	. . . 4 ⊢ ([,) Fn (ℝ* × ℝ*) → (([,) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ) ↔ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)))
6	3, 4, 5	mp2b 10	. . 3 ⊢ (([,) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ) ↔ (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
7	1, 6	mpbir 231	. 2 ⊢ ([,) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ)
8		eqid 2737	. . . . 5 ⊢ ([,) ↾ (ℝ × ℝ)) = ([,) ↾ (ℝ × ℝ))
9	8	icorempo 37352	. . . 4 ⊢ ([,) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
10	9	rneqi 5948	. . 3 ⊢ ran ([,) ↾ (ℝ × ℝ)) = ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
11		ssrab2 4080	. . . . . 6 ⊢ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ⊆ ℝ
12		reex 11246	. . . . . . 7 ⊢ ℝ ∈ V
13	12	elpw2 5334	. . . . . 6 ⊢ ({𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ↔ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ⊆ ℝ)
14	11, 13	mpbir 231	. . . . 5 ⊢ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ
15	14	rgen2w 3066	. . . 4 ⊢ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ
16		eqid 2737	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
17	16	rnmpo 7566	. . . . . . 7 ⊢ ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) = {𝑙 ∣ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}}
18	17	eqabri 2885	. . . . . 6 ⊢ (𝑙 ∈ ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
19		simpl 482	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ)
20		simpr 484	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
21	19, 20	r19.29d2r 3140	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ({𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}))
22		eleq1 2829	. . . . . . . . . . 11 ⊢ (𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} → (𝑙 ∈ 𝒫 ℝ ↔ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ))
23	22	biimparc 479	. . . . . . . . . 10 ⊢ (({𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) → 𝑙 ∈ 𝒫 ℝ)
24	23	a1i 11	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (({𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) → 𝑙 ∈ 𝒫 ℝ))
25	24	rexlimivv 3201	. . . . . . . 8 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ({𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) → 𝑙 ∈ 𝒫 ℝ)
26	21, 25	syl 17	. . . . . . 7 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) → 𝑙 ∈ 𝒫 ℝ)
27	26	ex 412	. . . . . 6 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} → 𝑙 ∈ 𝒫 ℝ))
28	18, 27	biimtrid 242	. . . . 5 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ → (𝑙 ∈ ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) → 𝑙 ∈ 𝒫 ℝ))
29	28	ssrdv 3989	. . . 4 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ → ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) ⊆ 𝒫 ℝ)
30	15, 29	ax-mp 5	. . 3 ⊢ ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) ⊆ 𝒫 ℝ
31	10, 30	eqsstri 4030	. 2 ⊢ ran ([,) ↾ (ℝ × ℝ)) ⊆ 𝒫 ℝ
32		df-f 6565	. 2 ⊢ (([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ ↔ (([,) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ) ∧ ran ([,) ↾ (ℝ × ℝ)) ⊆ 𝒫 ℝ))
33	7, 31, 32	mpbir2an 711	1 ⊢ ([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  {crab 3436   ⊆ wss 3951  𝒫 cpw 4600   class class class wbr 5143   × cxp 5683  ran crn 5686   ↾ cres 5687   Fn wfn 6556  ⟶wf 6557   ∈ cmpo 7433  ℝcr 11154  ℝ^*cxr 11294   < clt 11295   ≤ cle 11296  [,)cico 13389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-pre-lttri 11229  ax-pre-lttrn 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-ico 13393

This theorem is referenced by:  icoreelrnab  37355  icoreunrn  37360