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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 2822 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅)) | |
2 | infeq1 8928 | . . . 4 ⊢ (𝑥 = 𝐴 → inf(𝑥, ℝ*, < ) = inf(𝐴, ℝ*, < )) | |
3 | supeq1 8897 | . . . 4 ⊢ (𝑥 = 𝐴 → sup(𝑥, ℝ*, < ) = sup(𝐴, ℝ*, < )) | |
4 | 2, 3 | opeq12d 4803 | . . 3 ⊢ (𝑥 = 𝐴 → 〈inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )〉 = 〈inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )〉) |
5 | 1, 4 | ifbieq2d 4488 | . 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 5347 | . . 3 ⊢ 〈0, 0〉 ∈ V | |
8 | opex 5347 | . . 3 ⊢ 〈inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )〉 ∈ V | |
9 | 7, 8 | ifex 4511 | . 2 ⊢ if(𝐴 = ∅, 〈0, 0〉, 〈inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )〉) ∈ V |
10 | 5, 6, 9 | fvmpt 6761 | 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 1528 ∈ wcel 2105 ∅c0 4288 ifcif 4463 〈cop 4563 ↦ cmpt 5137 ran crn 5549 ‘cfv 6348 supcsup 8892 infcinf 8893 0cc0 10525 ℝ*cxr 10662 < clt 10663 (,)cioo 12726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-sup 8894 df-inf 8895 |
This theorem is referenced by: ioorinv2 24103 ioorinv 24104 ioorcl 24105 |
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