Theorem icoreresf 35523

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 11020	. . 3 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
2		df-ico 13085	. . . . 5 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
3	2	ixxf 13089	. . . 4 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
4		ffn 6600	. . . 4 ⊢ ([,):(ℝ* × ℝ*)⟶𝒫 ℝ* → [,) Fn (ℝ* × ℝ*))
5		fnssresb 6554	. . . 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 230	. 2 ⊢ ([,) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ)
8		eqid 2738	. . . . 5 ⊢ ([,) ↾ (ℝ × ℝ)) = ([,) ↾ (ℝ × ℝ))
9	8	icorempo 35522	. . . 4 ⊢ ([,) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
10	9	rneqi 5846	. . 3 ⊢ ran ([,) ↾ (ℝ × ℝ)) = ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
11		ssrab2 4013	. . . . . 6 ⊢ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ⊆ ℝ
12		reex 10962	. . . . . . 7 ⊢ ℝ ∈ V
13	12	elpw2 5269	. . . . . 6 ⊢ ({𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ↔ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ⊆ ℝ)
14	11, 13	mpbir 230	. . . . 5 ⊢ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ
15	14	rgen2w 3077	. . . 4 ⊢ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ
16		eqid 2738	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
17	16	rnmpo 7407	. . . . . . 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	abeq2i 2875	. . . . . 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 483	. . . . . . . . 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 485	. . . . . . . . 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 3264	. . . . . . . 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 2826	. . . . . . . . . . 11 ⊢ (𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} → (𝑙 ∈ 𝒫 ℝ ↔ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ))
23	22	biimparc 480	. . . . . . . . . 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 3221	. . . . . . . 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 413	. . . . . 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	syl5bi 241	. . . . 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 3927	. . . 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 3955	. 2 ⊢ ran ([,) ↾ (ℝ × ℝ)) ⊆ 𝒫 ℝ
32		df-f 6437	. 2 ⊢ (([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ ↔ (([,) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ) ∧ ran ([,) ↾ (ℝ × ℝ)) ⊆ 𝒫 ℝ))
33	7, 31, 32	mpbir2an 708	1 ⊢ ([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  {crab 3068   ⊆ wss 3887  𝒫 cpw 4533   class class class wbr 5074   × cxp 5587  ran crn 5590   ↾ cres 5591   Fn wfn 6428  ⟶wf 6429   ∈ cmpo 7277  ℝcr 10870  ℝ^*cxr 11008   < clt 11009   ≤ cle 11010  [,)cico 13081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-pre-lttri 10945  ax-pre-lttrn 10946

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-ico 13085

This theorem is referenced by:  icoreelrnab  35525  icoreunrn  35530