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 5962 | . . . 4 ⊢ (𝜑 → Fun 𝐹) |
3 | 1 | frnd 38421 | . . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
4 | fimacnvinrn2 6257 | . . . 4 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ ℝ) → (◡𝐹 “ (-∞[,)𝐵)) = (◡𝐹 “ ((-∞[,)𝐵) ∩ ℝ))) | |
5 | 2, 3, 4 | syl2anc 691 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (-∞[,)𝐵)) = (◡𝐹 “ ((-∞[,)𝐵) ∩ ℝ))) |
6 | preimaioomnf.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
7 | 6 | icomnfinre 38626 | . . . 4 ⊢ (𝜑 → ((-∞[,)𝐵) ∩ ℝ) = (-∞(,)𝐵)) |
8 | 7 | imaeq2d 5385 | . . 3 ⊢ (𝜑 → (◡𝐹 “ ((-∞[,)𝐵) ∩ ℝ)) = (◡𝐹 “ (-∞(,)𝐵))) |
9 | 5, 8 | eqtr2d 2645 | . 2 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝐵)) = (◡𝐹 “ (-∞[,)𝐵))) |
10 | 1 | frexr 38545 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
11 | 10, 6 | preimaicomnf 39599 | . 2 ⊢ (𝜑 → (◡𝐹 “ (-∞[,)𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐵}) |
12 | 9, 11 | eqtrd 2644 | 1 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {crab 2900 ∩ cin 3539 ⊆ wss 3540 class class class wbr 4583 ◡ccnv 5037 ran crn 5039 “ cima 5041 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 -∞cmnf 9951 ℝ*cxr 9952 < clt 9953 (,)cioo 12046 [,)cico 12048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-ioo 12050 df-ico 12052 |
This theorem is referenced by: issmflem 39613 mbfresmf 39626 smfres 39675 smfco 39687 |
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