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Mirrors > Home > MPE Home > Th. List > ioorval | Structured version Visualization version GIF version |
Description: Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.) |
Ref | Expression |
---|---|
ioorf.1 | ⊢ 𝐹 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, 〈0, 0〉, 〈inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )〉)) |
Ref | Expression |
---|---|
ioorval | ⊢ (𝐴 ∈ ran (,) → (𝐹‘𝐴) = if(𝐴 = ∅, 〈0, 0〉, 〈inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2614 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅)) | |
2 | infeq1 8265 | . . . 4 ⊢ (𝑥 = 𝐴 → inf(𝑥, ℝ*, < ) = inf(𝐴, ℝ*, < )) | |
3 | supeq1 8234 | . . . 4 ⊢ (𝑥 = 𝐴 → sup(𝑥, ℝ*, < ) = sup(𝐴, ℝ*, < )) | |
4 | 2, 3 | opeq12d 4348 | . . 3 ⊢ (𝑥 = 𝐴 → 〈inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )〉 = 〈inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )〉) |
5 | 1, 4 | ifbieq2d 4061 | . 2 ⊢ (𝑥 = 𝐴 → if(𝑥 = ∅, 〈0, 0〉, 〈inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )〉) = if(𝐴 = ∅, 〈0, 0〉, 〈inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )〉)) |
6 | ioorf.1 | . 2 ⊢ 𝐹 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, 〈0, 0〉, 〈inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )〉)) | |
7 | opex 4859 | . . 3 ⊢ 〈0, 0〉 ∈ V | |
8 | opex 4859 | . . 3 ⊢ 〈inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )〉 ∈ V | |
9 | 7, 8 | ifex 4106 | . 2 ⊢ if(𝐴 = ∅, 〈0, 0〉, 〈inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )〉) ∈ V |
10 | 5, 6, 9 | fvmpt 6191 | 1 ⊢ (𝐴 ∈ ran (,) → (𝐹‘𝐴) = if(𝐴 = ∅, 〈0, 0〉, 〈inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∅c0 3874 ifcif 4036 〈cop 4131 ↦ cmpt 4643 ran crn 5039 ‘cfv 5804 supcsup 8229 infcinf 8230 0cc0 9815 ℝ*cxr 9952 < clt 9953 (,)cioo 12046 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-sup 8231 df-inf 8232 |
This theorem is referenced by: ioorinv2 23149 ioorinv 23150 ioorcl 23151 |
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