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Mirrors > Home > MPE Home > Th. List > Mathboxes > rightval | Structured version Visualization version GIF version |
Description: The value of the right options function. (Contributed by Scott Fenton, 6-Aug-2024.) |
Ref | Expression |
---|---|
rightval | ⊢ (𝐴 ∈ No → ( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2fveq3 6668 | . . 3 ⊢ (𝑦 = 𝐴 → ( O ‘( bday ‘𝑦)) = ( O ‘( bday ‘𝐴))) | |
2 | breq1 5039 | . . 3 ⊢ (𝑦 = 𝐴 → (𝑦 <s 𝑥 ↔ 𝐴 <s 𝑥)) | |
3 | 1, 2 | rabeqbidv 3398 | . 2 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ ( O ‘( bday ‘𝑦)) ∣ 𝑦 <s 𝑥} = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥}) |
4 | df-right 33629 | . 2 ⊢ R = (𝑦 ∈ No ↦ {𝑥 ∈ ( O ‘( bday ‘𝑦)) ∣ 𝑦 <s 𝑥}) | |
5 | fvex 6676 | . . 3 ⊢ ( O ‘( bday ‘𝐴)) ∈ V | |
6 | 5 | rabex 5206 | . 2 ⊢ {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥} ∈ V |
7 | 3, 4, 6 | fvmpt 6764 | 1 ⊢ (𝐴 ∈ No → ( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {crab 3074 class class class wbr 5036 ‘cfv 6340 No csur 33440 <s cslt 33441 bday cbday 33442 O cold 33621 R cright 33624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5173 ax-nul 5180 ax-pr 5302 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ral 3075 df-rex 3076 df-rab 3079 df-v 3411 df-sbc 3699 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-br 5037 df-opab 5099 df-mpt 5117 df-id 5434 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-iota 6299 df-fun 6342 df-fv 6348 df-right 33629 |
This theorem is referenced by: ssltright 33645 rightssold 33653 lrold 33668 madebdaylemlrcut 33670 |
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