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Mirrors > Home > MPE Home > Th. List > Mathboxes > ind0 | Structured version Visualization version GIF version |
Description: Value of the indicator function where it is 0. (Contributed by Thierry Arnoux, 14-Aug-2017.) |
Ref | Expression |
---|---|
ind0 | ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ (𝑂 ∖ 𝐴)) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifi 4106 | . . 3 ⊢ (𝑋 ∈ (𝑂 ∖ 𝐴) → 𝑋 ∈ 𝑂) | |
2 | indfval 31279 | . . 3 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = if(𝑋 ∈ 𝐴, 1, 0)) | |
3 | 1, 2 | syl3an3 1161 | . 2 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ (𝑂 ∖ 𝐴)) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = if(𝑋 ∈ 𝐴, 1, 0)) |
4 | eldifn 4107 | . . . 4 ⊢ (𝑋 ∈ (𝑂 ∖ 𝐴) → ¬ 𝑋 ∈ 𝐴) | |
5 | 4 | 3ad2ant3 1131 | . . 3 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ (𝑂 ∖ 𝐴)) → ¬ 𝑋 ∈ 𝐴) |
6 | 5 | iffalsed 4481 | . 2 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ (𝑂 ∖ 𝐴)) → if(𝑋 ∈ 𝐴, 1, 0) = 0) |
7 | 3, 6 | eqtrd 2859 | 1 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ (𝑂 ∖ 𝐴)) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ∖ cdif 3936 ⊆ wss 3939 ifcif 4470 ‘cfv 6358 0cc0 10540 1c1 10541 𝟭cind 31273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-i2m1 10608 ax-1ne0 10609 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-ind 31274 |
This theorem is referenced by: indsum 31284 indsumin 31285 |
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