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Theorem hoissre 40527
Description: The projection of a half-open interval onto a single dimension is a subset of . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypothesis
Ref Expression
hoissre.1 (𝜑𝐼:𝑋⟶(ℝ × ℝ))
Assertion
Ref Expression
hoissre ((𝜑𝑘𝑋) → (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ)
Distinct variable group:   𝑘,𝑋
Allowed substitution hints:   𝜑(𝑘)   𝐼(𝑘)

Proof of Theorem hoissre
StepHypRef Expression
1 hoissre.1 . . . 4 (𝜑𝐼:𝑋⟶(ℝ × ℝ))
21adantr 481 . . 3 ((𝜑𝑘𝑋) → 𝐼:𝑋⟶(ℝ × ℝ))
3 simpr 477 . . 3 ((𝜑𝑘𝑋) → 𝑘𝑋)
42, 3fvovco 39203 . 2 ((𝜑𝑘𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((1st ‘(𝐼𝑘))[,)(2nd ‘(𝐼𝑘))))
51ffvelrnda 6357 . . . 4 ((𝜑𝑘𝑋) → (𝐼𝑘) ∈ (ℝ × ℝ))
6 xp1st 7195 . . . 4 ((𝐼𝑘) ∈ (ℝ × ℝ) → (1st ‘(𝐼𝑘)) ∈ ℝ)
75, 6syl 17 . . 3 ((𝜑𝑘𝑋) → (1st ‘(𝐼𝑘)) ∈ ℝ)
8 xp2nd 7196 . . . . 5 ((𝐼𝑘) ∈ (ℝ × ℝ) → (2nd ‘(𝐼𝑘)) ∈ ℝ)
95, 8syl 17 . . . 4 ((𝜑𝑘𝑋) → (2nd ‘(𝐼𝑘)) ∈ ℝ)
109rexrd 10086 . . 3 ((𝜑𝑘𝑋) → (2nd ‘(𝐼𝑘)) ∈ ℝ*)
11 icossre 12251 . . 3 (((1st ‘(𝐼𝑘)) ∈ ℝ ∧ (2nd ‘(𝐼𝑘)) ∈ ℝ*) → ((1st ‘(𝐼𝑘))[,)(2nd ‘(𝐼𝑘))) ⊆ ℝ)
127, 10, 11syl2anc 693 . 2 ((𝜑𝑘𝑋) → ((1st ‘(𝐼𝑘))[,)(2nd ‘(𝐼𝑘))) ⊆ ℝ)
134, 12eqsstrd 3637 1 ((𝜑𝑘𝑋) → (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1989  wss 3572   × cxp 5110  ccom 5116  wf 5882  cfv 5886  (class class class)co 6647  1st c1st 7163  2nd c2nd 7164  cr 9932  *cxr 10070  [,)cico 12174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-pre-lttri 10007  ax-pre-lttrn 10008
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-po 5033  df-so 5034  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-1st 7165  df-2nd 7166  df-er 7739  df-en 7953  df-dom 7954  df-sdom 7955  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-ico 12178
This theorem is referenced by:  hoissrrn  40532
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