Theorem icoreresf 37604

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 11189	. . 3 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
2		df-ico 13279	. . . . 5 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
3	2	ixxf 13283	. . . 4 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
4		ffn 6670	. . . 4 ⊢ ([,):(ℝ* × ℝ*)⟶𝒫 ℝ* → [,) Fn (ℝ* × ℝ*))
5		fnssresb 6622	. . . 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 37603	. . . 4 ⊢ ([,) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
10	9	rneqi 5894	. . 3 ⊢ ran ([,) ↾ (ℝ × ℝ)) = ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
11		ssrab2 4034	. . . . . 6 ⊢ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ⊆ ℝ
12		reex 11129	. . . . . . 7 ⊢ ℝ ∈ V
13	12	elpw2 5281	. . . . . 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 3057	. . . 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 7501	. . . . . . 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 2879	. . . . . 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 3125	. . . . . . . 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 2825	. . . . . . . . . . 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 3180	. . . . . . . 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 3941	. . . 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 3982	. 2 ⊢ ran ([,) ↾ (ℝ × ℝ)) ⊆ 𝒫 ℝ
32		df-f 6504	. 2 ⊢ (([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ ↔ (([,) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ) ∧ ran ([,) ↾ (ℝ × ℝ)) ⊆ 𝒫 ℝ))
33	7, 31, 32	mpbir2an 712	1 ⊢ ([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {crab 3401   ⊆ wss 3903  𝒫 cpw 4556   class class class wbr 5100   × cxp 5630  ran crn 5633   ↾ cres 5634   Fn wfn 6495  ⟶wf 6496   ∈ cmpo 7370  ℝcr 11037  ℝ^*cxr 11177   < clt 11178   ≤ cle 11179  [,)cico 13275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-pre-lttri 11112  ax-pre-lttrn 11113

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-po 5540  df-so 5541  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-ico 13279

This theorem is referenced by:  icoreelrnab  37606  icoreunrn  37611