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Theorem icorempo 34649
Description: Closed-below, open-above intervals of reals. (Contributed by ML, 26-Jul-2020.)
Hypothesis
Ref Expression
icorempo.1 𝐹 = ([,) ↾ (ℝ × ℝ))
Assertion
Ref Expression
icorempo 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
Distinct variable group:   𝑥,𝑦,𝑧
Allowed substitution hints:   𝐹(𝑥,𝑦,𝑧)

Proof of Theorem icorempo
StepHypRef Expression
1 icorempo.1 . 2 𝐹 = ([,) ↾ (ℝ × ℝ))
2 df-ico 12723 . . . 4 [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
32reseq1i 5825 . . 3 ([,) ↾ (ℝ × ℝ)) = ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}) ↾ (ℝ × ℝ))
4 ressxr 10663 . . . 4 ℝ ⊆ ℝ*
5 resmpo 7249 . . . 4 ((ℝ ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}))
64, 4, 5mp2an 690 . . 3 ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
73, 6eqtri 2843 . 2 ([,) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
8 nfv 1915 . . . 4 𝑧(𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)
9 nfrab1 3371 . . . 4 𝑧{𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}
10 nfrab1 3371 . . . 4 𝑧{𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}
11 rabid 3365 . . . . . . . 8 (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ↔ (𝑧 ∈ ℝ* ∧ (𝑥𝑧𝑧 < 𝑦)))
12 rexr 10665 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
13 nltmnf 12503 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ* → ¬ 𝑥 < -∞)
1412, 13syl 17 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℝ → ¬ 𝑥 < -∞)
15 renemnf 10668 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ → 𝑥 ≠ -∞)
1615neneqd 3011 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℝ → ¬ 𝑥 = -∞)
1714, 16jca 514 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ → (¬ 𝑥 < -∞ ∧ ¬ 𝑥 = -∞))
18 pm4.56 985 . . . . . . . . . . . . . . 15 ((¬ 𝑥 < -∞ ∧ ¬ 𝑥 = -∞) ↔ ¬ (𝑥 < -∞ ∨ 𝑥 = -∞))
1917, 18sylib 220 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → ¬ (𝑥 < -∞ ∨ 𝑥 = -∞))
20 mnfxr 10676 . . . . . . . . . . . . . . 15 -∞ ∈ ℝ*
21 xrleloe 12516 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (𝑥 ≤ -∞ ↔ (𝑥 < -∞ ∨ 𝑥 = -∞)))
2212, 20, 21sylancl 588 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → (𝑥 ≤ -∞ ↔ (𝑥 < -∞ ∨ 𝑥 = -∞)))
2319, 22mtbird 327 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → ¬ 𝑥 ≤ -∞)
24 breq2 5046 . . . . . . . . . . . . . 14 (𝑧 = -∞ → (𝑥𝑧𝑥 ≤ -∞))
2524notbid 320 . . . . . . . . . . . . 13 (𝑧 = -∞ → (¬ 𝑥𝑧 ↔ ¬ 𝑥 ≤ -∞))
2623, 25syl5ibrcom 249 . . . . . . . . . . . 12 (𝑥 ∈ ℝ → (𝑧 = -∞ → ¬ 𝑥𝑧))
2726con2d 136 . . . . . . . . . . 11 (𝑥 ∈ ℝ → (𝑥𝑧 → ¬ 𝑧 = -∞))
28 rexr 10665 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
29 pnfnlt 12502 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦)
30 breq1 5045 . . . . . . . . . . . . . . 15 (𝑧 = +∞ → (𝑧 < 𝑦 ↔ +∞ < 𝑦))
3130notbid 320 . . . . . . . . . . . . . 14 (𝑧 = +∞ → (¬ 𝑧 < 𝑦 ↔ ¬ +∞ < 𝑦))
3229, 31syl5ibrcom 249 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → (𝑧 = +∞ → ¬ 𝑧 < 𝑦))
3332con2d 136 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → (𝑧 < 𝑦 → ¬ 𝑧 = +∞))
3428, 33syl 17 . . . . . . . . . . 11 (𝑦 ∈ ℝ → (𝑧 < 𝑦 → ¬ 𝑧 = +∞))
3527, 34im2anan9 621 . . . . . . . . . 10 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥𝑧𝑧 < 𝑦) → (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞)))
3635anim2d 613 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 ∈ ℝ* ∧ (𝑥𝑧𝑧 < 𝑦)) → (𝑧 ∈ ℝ* ∧ (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞))))
37 renepnf 10667 . . . . . . . . . . . 12 (𝑧 ∈ ℝ → 𝑧 ≠ +∞)
3837neneqd 3011 . . . . . . . . . . 11 (𝑧 ∈ ℝ → ¬ 𝑧 = +∞)
3938pm4.71i 562 . . . . . . . . . 10 (𝑧 ∈ ℝ ↔ (𝑧 ∈ ℝ ∧ ¬ 𝑧 = +∞))
40 xrnemnf 12491 . . . . . . . . . . . 12 ((𝑧 ∈ ℝ*𝑧 ≠ -∞) ↔ (𝑧 ∈ ℝ ∨ 𝑧 = +∞))
4140anbi1i 625 . . . . . . . . . . 11 (((𝑧 ∈ ℝ*𝑧 ≠ -∞) ∧ ¬ 𝑧 = +∞) ↔ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∧ ¬ 𝑧 = +∞))
42 df-ne 3007 . . . . . . . . . . . . 13 (𝑧 ≠ -∞ ↔ ¬ 𝑧 = -∞)
4342anbi2i 624 . . . . . . . . . . . 12 ((𝑧 ∈ ℝ*𝑧 ≠ -∞) ↔ (𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞))
4443anbi1i 625 . . . . . . . . . . 11 (((𝑧 ∈ ℝ*𝑧 ≠ -∞) ∧ ¬ 𝑧 = +∞) ↔ ((𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞) ∧ ¬ 𝑧 = +∞))
45 pm5.61 997 . . . . . . . . . . 11 (((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∧ ¬ 𝑧 = +∞) ↔ (𝑧 ∈ ℝ ∧ ¬ 𝑧 = +∞))
4641, 44, 453bitr3i 303 . . . . . . . . . 10 (((𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞) ∧ ¬ 𝑧 = +∞) ↔ (𝑧 ∈ ℝ ∧ ¬ 𝑧 = +∞))
47 anass 471 . . . . . . . . . 10 (((𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞) ∧ ¬ 𝑧 = +∞) ↔ (𝑧 ∈ ℝ* ∧ (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞)))
4839, 46, 473bitr2ri 302 . . . . . . . . 9 ((𝑧 ∈ ℝ* ∧ (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞)) ↔ 𝑧 ∈ ℝ)
4936, 48syl6ib 253 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 ∈ ℝ* ∧ (𝑥𝑧𝑧 < 𝑦)) → 𝑧 ∈ ℝ))
5011, 49syl5bi 244 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} → 𝑧 ∈ ℝ))
5111simprbi 499 . . . . . . . 8 (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} → (𝑥𝑧𝑧 < 𝑦))
5251a1i 11 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} → (𝑥𝑧𝑧 < 𝑦)))
5350, 52jcad 515 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} → (𝑧 ∈ ℝ ∧ (𝑥𝑧𝑧 < 𝑦))))
54 rabid 3365 . . . . . 6 (𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ↔ (𝑧 ∈ ℝ ∧ (𝑥𝑧𝑧 < 𝑦)))
5553, 54syl6ibr 254 . . . . 5 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} → 𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}))
56 rabss2 4033 . . . . . . 7 (ℝ ⊆ ℝ* → {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ⊆ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
574, 56ax-mp 5 . . . . . 6 {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ⊆ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}
5857sseli 3942 . . . . 5 (𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} → 𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
5955, 58impbid1 227 . . . 4 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ↔ 𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}))
608, 9, 10, 59eqrd 3965 . . 3 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
6160mpoeq3ia 7209 . 2 (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
621, 7, 613eqtri 2847 1 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843   = wceq 1537  wcel 2114  wne 3006  {crab 3129  wss 3913   class class class wbr 5042   × cxp 5529  cres 5533  cmpo 7135  cr 10514  +∞cpnf 10650  -∞cmnf 10651  *cxr 10652   < clt 10653  cle 10654  [,)cico 12719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-cnex 10571  ax-resscn 10572  ax-pre-lttri 10589  ax-pre-lttrn 10590
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-po 5450  df-so 5451  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-oprab 7137  df-mpo 7138  df-er 8267  df-en 8488  df-dom 8489  df-sdom 8490  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-ico 12723
This theorem is referenced by:  icoreresf  34650  icoreval  34651
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