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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > indfval | Structured version Visualization version GIF version |
Description: Value of the indicator function. (Contributed by Thierry Arnoux, 13-Aug-2017.) |
Ref | Expression |
---|---|
indfval | ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = if(𝑋 ∈ 𝐴, 1, 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indval 32948 | . . 3 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0))) | |
2 | 1 | 3adant3 1133 | . 2 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0))) |
3 | simpr 486 | . . . 4 ⊢ (((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
4 | 3 | eleq1d 2819 | . . 3 ⊢ (((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂) ∧ 𝑥 = 𝑋) → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴)) |
5 | 4 | ifbid 4549 | . 2 ⊢ (((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂) ∧ 𝑥 = 𝑋) → if(𝑥 ∈ 𝐴, 1, 0) = if(𝑋 ∈ 𝐴, 1, 0)) |
6 | simp3 1139 | . 2 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂) → 𝑋 ∈ 𝑂) | |
7 | 1re 11209 | . . . 4 ⊢ 1 ∈ ℝ | |
8 | 0re 11211 | . . . 4 ⊢ 0 ∈ ℝ | |
9 | 7, 8 | ifcli 4573 | . . 3 ⊢ if(𝑋 ∈ 𝐴, 1, 0) ∈ ℝ |
10 | 9 | a1i 11 | . 2 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂) → if(𝑋 ∈ 𝐴, 1, 0) ∈ ℝ) |
11 | 2, 5, 6, 10 | fvmptd 7000 | 1 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = if(𝑋 ∈ 𝐴, 1, 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ⊆ wss 3946 ifcif 4526 ↦ cmpt 5229 ‘cfv 6539 ℝcr 11104 0cc0 11105 1c1 11106 𝟭cind 32945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-i2m1 11173 ax-1ne0 11174 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-ov 7406 df-ind 32946 |
This theorem is referenced by: ind1 32952 ind0 32953 ind1a 32954 eulerpartlemgvv 33312 |
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