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Mathbox for Glauco Siliprandi |
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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 6295 | . . . 4 ⊢ (𝜑 → Fun 𝐹) |
3 | 1 | frnd 6298 | . . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
4 | fimacnvinrn2 6613 | . . . 4 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ ℝ) → (◡𝐹 “ (-∞[,)𝐵)) = (◡𝐹 “ ((-∞[,)𝐵) ∩ ℝ))) | |
5 | 2, 3, 4 | syl2anc 579 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (-∞[,)𝐵)) = (◡𝐹 “ ((-∞[,)𝐵) ∩ ℝ))) |
6 | preimaioomnf.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
7 | 6 | icomnfinre 40669 | . . . 4 ⊢ (𝜑 → ((-∞[,)𝐵) ∩ ℝ) = (-∞(,)𝐵)) |
8 | 7 | imaeq2d 5720 | . . 3 ⊢ (𝜑 → (◡𝐹 “ ((-∞[,)𝐵) ∩ ℝ)) = (◡𝐹 “ (-∞(,)𝐵))) |
9 | 5, 8 | eqtr2d 2814 | . 2 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝐵)) = (◡𝐹 “ (-∞[,)𝐵))) |
10 | 1 | frexr 40494 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
11 | 10, 6 | preimaicomnf 41831 | . 2 ⊢ (𝜑 → (◡𝐹 “ (-∞[,)𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐵}) |
12 | 9, 11 | eqtrd 2813 | 1 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 {crab 3093 ∩ cin 3790 ⊆ wss 3791 class class class wbr 4886 ◡ccnv 5354 ran crn 5356 “ cima 5358 Fun wfun 6129 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ℝcr 10271 -∞cmnf 10409 ℝ*cxr 10410 < clt 10411 (,)cioo 12487 [,)cico 12489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-pre-lttri 10346 ax-pre-lttrn 10347 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-ioo 12491 df-ico 12493 |
This theorem is referenced by: issmflem 41845 mbfresmf 41857 smfres 41906 smfco 41918 |
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