![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > preimaioomnf | Structured version Visualization version GIF version |
Description: Preimage of an open interval, unbounded below. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
preimaioomnf.1 | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
preimaioomnf.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
Ref | Expression |
---|---|
preimaioomnf | ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preimaioomnf.1 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
2 | 1 | ffund 6718 | . . . 4 ⊢ (𝜑 → Fun 𝐹) |
3 | 1 | frnd 6722 | . . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
4 | fimacnvinrn2 7070 | . . . 4 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ ℝ) → (◡𝐹 “ (-∞[,)𝐵)) = (◡𝐹 “ ((-∞[,)𝐵) ∩ ℝ))) | |
5 | 2, 3, 4 | syl2anc 585 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (-∞[,)𝐵)) = (◡𝐹 “ ((-∞[,)𝐵) ∩ ℝ))) |
6 | preimaioomnf.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
7 | 6 | icomnfinre 44200 | . . . 4 ⊢ (𝜑 → ((-∞[,)𝐵) ∩ ℝ) = (-∞(,)𝐵)) |
8 | 7 | imaeq2d 6057 | . . 3 ⊢ (𝜑 → (◡𝐹 “ ((-∞[,)𝐵) ∩ ℝ)) = (◡𝐹 “ (-∞(,)𝐵))) |
9 | 5, 8 | eqtr2d 2774 | . 2 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝐵)) = (◡𝐹 “ (-∞[,)𝐵))) |
10 | 1 | frexr 44030 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
11 | 10, 6 | preimaicomnf 45362 | . 2 ⊢ (𝜑 → (◡𝐹 “ (-∞[,)𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐵}) |
12 | 9, 11 | eqtrd 2773 | 1 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {crab 3433 ∩ cin 3946 ⊆ wss 3947 class class class wbr 5147 ◡ccnv 5674 ran crn 5676 “ cima 5678 Fun wfun 6534 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 ℝcr 11105 -∞cmnf 11242 ℝ*cxr 11243 < clt 11244 (,)cioo 13320 [,)cico 13322 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-pre-lttri 11180 ax-pre-lttrn 11181 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7970 df-2nd 7971 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-ioo 13324 df-ico 13326 |
This theorem is referenced by: issmflem 45378 mbfresmf 45390 smfres 45441 smfco 45453 |
Copyright terms: Public domain | W3C validator |