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Mirrors > Home > MPE Home > Th. List > Mathboxes > lrrecpred | Structured version Visualization version GIF version |
Description: Finally, we calculate the value of the predecessor class over 𝑅. (Contributed by Scott Fenton, 19-Aug-2024.) |
Ref | Expression |
---|---|
lrrec.1 | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} |
Ref | Expression |
---|---|
lrrecpred | ⊢ (𝐴 ∈ No → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfpred3g 6141 | . 2 ⊢ (𝐴 ∈ No → Pred(𝑅, No , 𝐴) = {𝑏 ∈ No ∣ 𝑏𝑅𝐴}) | |
2 | lrrec.1 | . . . . . 6 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} | |
3 | 2 | lrrecval 33670 | . . . . 5 ⊢ ((𝑏 ∈ No ∧ 𝐴 ∈ No ) → (𝑏𝑅𝐴 ↔ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))) |
4 | 3 | ancoms 462 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑏 ∈ No ) → (𝑏𝑅𝐴 ↔ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))) |
5 | 4 | rabbidva 3390 | . . 3 ⊢ (𝐴 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝐴} = {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))}) |
6 | dfrab2 4215 | . . . 4 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} = ({𝑏 ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} ∩ No ) | |
7 | abid2 2894 | . . . . 5 ⊢ {𝑏 ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} = (( L ‘𝐴) ∪ ( R ‘𝐴)) | |
8 | 7 | ineq1i 4115 | . . . 4 ⊢ ({𝑏 ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} ∩ No ) = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No ) |
9 | 6, 8 | eqtri 2781 | . . 3 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No ) |
10 | 5, 9 | eqtrdi 2809 | . 2 ⊢ (𝐴 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝐴} = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No )) |
11 | leftssold 33644 | . . . . 5 ⊢ (𝐴 ∈ No → ( L ‘𝐴) ⊆ ( O ‘( bday ‘𝐴))) | |
12 | bdayelon 33560 | . . . . . 6 ⊢ ( bday ‘𝐴) ∈ On | |
13 | oldf 33627 | . . . . . . . 8 ⊢ O :On⟶𝒫 No | |
14 | 13 | ffvelrni 6846 | . . . . . . 7 ⊢ (( bday ‘𝐴) ∈ On → ( O ‘( bday ‘𝐴)) ∈ 𝒫 No ) |
15 | 14 | elpwid 4508 | . . . . . 6 ⊢ (( bday ‘𝐴) ∈ On → ( O ‘( bday ‘𝐴)) ⊆ No ) |
16 | 12, 15 | ax-mp 5 | . . . . 5 ⊢ ( O ‘( bday ‘𝐴)) ⊆ No |
17 | 11, 16 | sstrdi 3906 | . . . 4 ⊢ (𝐴 ∈ No → ( L ‘𝐴) ⊆ No ) |
18 | rightssold 33645 | . . . . 5 ⊢ (𝐴 ∈ No → ( R ‘𝐴) ⊆ ( O ‘( bday ‘𝐴))) | |
19 | 18, 16 | sstrdi 3906 | . . . 4 ⊢ (𝐴 ∈ No → ( R ‘𝐴) ⊆ No ) |
20 | 17, 19 | unssd 4093 | . . 3 ⊢ (𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) ⊆ No ) |
21 | df-ss 3877 | . . 3 ⊢ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ⊆ No ↔ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No ) = (( L ‘𝐴) ∪ ( R ‘𝐴))) | |
22 | 20, 21 | sylib 221 | . 2 ⊢ (𝐴 ∈ No → ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No ) = (( L ‘𝐴) ∪ ( R ‘𝐴))) |
23 | 1, 10, 22 | 3eqtrd 2797 | 1 ⊢ (𝐴 ∈ No → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 {cab 2735 {crab 3074 ∪ cun 3858 ∩ cin 3859 ⊆ wss 3860 𝒫 cpw 4497 class class class wbr 5035 {copab 5097 Predcpred 6129 Oncon0 6173 ‘cfv 6339 No csur 33432 bday cbday 33434 O cold 33613 L cleft 33615 R cright 33616 |
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-rep 5159 ax-sep 5172 ax-nul 5179 ax-pow 5237 ax-pr 5301 ax-un 7464 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 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-ne 2952 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-br 5036 df-opab 5098 df-mpt 5116 df-tr 5142 df-id 5433 df-eprel 5438 df-po 5446 df-so 5447 df-fr 5486 df-we 5488 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-res 5539 df-ima 5540 df-pred 6130 df-ord 6176 df-on 6177 df-suc 6179 df-iota 6298 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-wrecs 7962 df-recs 8023 df-1o 8117 df-2o 8118 df-no 33435 df-slt 33436 df-bday 33437 df-sslt 33565 df-scut 33567 df-made 33617 df-old 33618 df-left 33620 df-right 33621 |
This theorem is referenced by: noinds 33676 norecov 33678 noxpordpred 33684 no2indslem 33685 no3indslem 33689 |
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