Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2827 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅)) | |
2 | infeq1 8942 | . . . 4 ⊢ (𝑥 = 𝐴 → inf(𝑥, ℝ*, < ) = inf(𝐴, ℝ*, < )) | |
3 | supeq1 8911 | . . . 4 ⊢ (𝑥 = 𝐴 → sup(𝑥, ℝ*, < ) = sup(𝐴, ℝ*, < )) | |
4 | 2, 3 | opeq12d 4813 | . . 3 ⊢ (𝑥 = 𝐴 → 〈inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )〉 = 〈inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )〉) |
5 | 1, 4 | ifbieq2d 4494 | . 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 5358 | . . 3 ⊢ 〈0, 0〉 ∈ V | |
8 | opex 5358 | . . 3 ⊢ 〈inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )〉 ∈ V | |
9 | 7, 8 | ifex 4517 | . 2 ⊢ if(𝐴 = ∅, 〈0, 0〉, 〈inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )〉) ∈ V |
10 | 5, 6, 9 | fvmpt 6770 | 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 1537 ∈ wcel 2114 ∅c0 4293 ifcif 4469 〈cop 4575 ↦ cmpt 5148 ran crn 5558 ‘cfv 6357 supcsup 8906 infcinf 8907 0cc0 10539 ℝ*cxr 10676 < clt 10677 (,)cioo 12741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-sup 8908 df-inf 8909 |
This theorem is referenced by: ioorinv2 24178 ioorinv 24179 ioorcl 24180 |
Copyright terms: Public domain | W3C validator |