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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 6491 | . . . 4 ⊢ (𝜑 → Fun 𝐹) |
3 | 1 | frnd 6494 | . . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
4 | fimacnvinrn2 6818 | . . . 4 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ ℝ) → (◡𝐹 “ (-∞[,)𝐵)) = (◡𝐹 “ ((-∞[,)𝐵) ∩ ℝ))) | |
5 | 2, 3, 4 | syl2anc 587 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (-∞[,)𝐵)) = (◡𝐹 “ ((-∞[,)𝐵) ∩ ℝ))) |
6 | preimaioomnf.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
7 | 6 | icomnfinre 42189 | . . . 4 ⊢ (𝜑 → ((-∞[,)𝐵) ∩ ℝ) = (-∞(,)𝐵)) |
8 | 7 | imaeq2d 5896 | . . 3 ⊢ (𝜑 → (◡𝐹 “ ((-∞[,)𝐵) ∩ ℝ)) = (◡𝐹 “ (-∞(,)𝐵))) |
9 | 5, 8 | eqtr2d 2834 | . 2 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝐵)) = (◡𝐹 “ (-∞[,)𝐵))) |
10 | 1 | frexr 42019 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
11 | 10, 6 | preimaicomnf 43347 | . 2 ⊢ (𝜑 → (◡𝐹 “ (-∞[,)𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐵}) |
12 | 9, 11 | eqtrd 2833 | 1 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {crab 3110 ∩ cin 3880 ⊆ wss 3881 class class class wbr 5030 ◡ccnv 5518 ran crn 5520 “ cima 5522 Fun wfun 6318 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 -∞cmnf 10662 ℝ*cxr 10663 < clt 10664 (,)cioo 12726 [,)cico 12728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-ioo 12730 df-ico 12732 |
This theorem is referenced by: issmflem 43361 mbfresmf 43373 smfres 43422 smfco 43434 |
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