Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > preimaicomnf | Structured version Visualization version GIF version |
Description: Preimage of an open interval, unbounded below. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
preimaicomnf.1 | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
preimaicomnf.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
Ref | Expression |
---|---|
preimaicomnf | ⊢ (𝜑 → (◡𝐹 “ (-∞[,)𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preimaicomnf.1 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) | |
2 | 1 | ffnd 6515 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
3 | fncnvima2 6831 | . . 3 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ (-∞[,)𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ (-∞[,)𝐵)}) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝜑 → (◡𝐹 “ (-∞[,)𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ (-∞[,)𝐵)}) |
5 | mnfxr 10698 | . . . . . . 7 ⊢ -∞ ∈ ℝ* | |
6 | 5 | a1i 11 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) ∈ (-∞[,)𝐵)) → -∞ ∈ ℝ*) |
7 | preimaicomnf.2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
8 | 7 | ad2antrr 724 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) ∈ (-∞[,)𝐵)) → 𝐵 ∈ ℝ*) |
9 | simpr 487 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) ∈ (-∞[,)𝐵)) → (𝐹‘𝑥) ∈ (-∞[,)𝐵)) | |
10 | icoltub 41804 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝐹‘𝑥) ∈ (-∞[,)𝐵)) → (𝐹‘𝑥) < 𝐵) | |
11 | 6, 8, 9, 10 | syl3anc 1367 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) ∈ (-∞[,)𝐵)) → (𝐹‘𝑥) < 𝐵) |
12 | 11 | ex 415 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) ∈ (-∞[,)𝐵) → (𝐹‘𝑥) < 𝐵)) |
13 | 5 | a1i 11 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) < 𝐵) → -∞ ∈ ℝ*) |
14 | 7 | ad2antrr 724 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) < 𝐵) → 𝐵 ∈ ℝ*) |
15 | 1 | ffvelrnda 6851 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℝ*) |
16 | 15 | adantr 483 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) < 𝐵) → (𝐹‘𝑥) ∈ ℝ*) |
17 | 15 | mnfled 41680 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -∞ ≤ (𝐹‘𝑥)) |
18 | 17 | adantr 483 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) < 𝐵) → -∞ ≤ (𝐹‘𝑥)) |
19 | simpr 487 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) < 𝐵) → (𝐹‘𝑥) < 𝐵) | |
20 | 13, 14, 16, 18, 19 | elicod 12788 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) < 𝐵) → (𝐹‘𝑥) ∈ (-∞[,)𝐵)) |
21 | 20 | ex 415 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) < 𝐵 → (𝐹‘𝑥) ∈ (-∞[,)𝐵))) |
22 | 12, 21 | impbid 214 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) ∈ (-∞[,)𝐵) ↔ (𝐹‘𝑥) < 𝐵)) |
23 | 22 | rabbidva 3478 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ (-∞[,)𝐵)} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐵}) |
24 | 4, 23 | eqtrd 2856 | 1 ⊢ (𝜑 → (◡𝐹 “ (-∞[,)𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3142 class class class wbr 5066 ◡ccnv 5554 “ cima 5558 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 -∞cmnf 10673 ℝ*cxr 10674 < clt 10675 ≤ cle 10676 [,)cico 12741 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-ico 12745 |
This theorem is referenced by: preimaioomnf 43017 |
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