Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
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 6225 | . 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 34137 | . . . . 5 ⊢ ((𝑏 ∈ No ∧ 𝐴 ∈ No ) → (𝑏𝑅𝐴 ↔ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))) |
4 | 3 | ancoms 460 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑏 ∈ No ) → (𝑏𝑅𝐴 ↔ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))) |
5 | 4 | rabbidva 3420 | . . 3 ⊢ (𝐴 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝐴} = {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))}) |
6 | dfrab2 4250 | . . . 4 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} = ({𝑏 ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} ∩ No ) | |
7 | abid2 2880 | . . . . 5 ⊢ {𝑏 ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} = (( L ‘𝐴) ∪ ( R ‘𝐴)) | |
8 | 7 | ineq1i 4148 | . . . 4 ⊢ ({𝑏 ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} ∩ No ) = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No ) |
9 | 6, 8 | eqtri 2764 | . . 3 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No ) |
10 | 5, 9 | eqtrdi 2792 | . 2 ⊢ (𝐴 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝐴} = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No )) |
11 | leftssno 34104 | . . . . 5 ⊢ ( L ‘𝐴) ⊆ No | |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ No → ( L ‘𝐴) ⊆ No ) |
13 | rightssno 34105 | . . . . 5 ⊢ ( R ‘𝐴) ⊆ No | |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ No → ( R ‘𝐴) ⊆ No ) |
15 | 12, 14 | unssd 4126 | . . 3 ⊢ (𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) ⊆ No ) |
16 | df-ss 3909 | . . 3 ⊢ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ⊆ No ↔ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No ) = (( L ‘𝐴) ∪ ( R ‘𝐴))) | |
17 | 15, 16 | sylib 217 | . 2 ⊢ (𝐴 ∈ No → ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No ) = (( L ‘𝐴) ∪ ( R ‘𝐴))) |
18 | 1, 10, 17 | 3eqtrd 2780 | 1 ⊢ (𝐴 ∈ No → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2104 {cab 2713 {crab 3284 ∪ cun 3890 ∩ cin 3891 ⊆ wss 3892 class class class wbr 5081 {copab 5143 Predcpred 6212 ‘cfv 6454 No csur 33884 L cleft 34070 R cright 34071 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5496 df-eprel 5502 df-po 5510 df-so 5511 df-fr 5551 df-we 5553 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-pred 6213 df-ord 6280 df-on 6281 df-suc 6283 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-riota 7260 df-ov 7306 df-oprab 7307 df-mpo 7308 df-2nd 7860 df-frecs 8124 df-wrecs 8155 df-recs 8229 df-1o 8324 df-2o 8325 df-no 33887 df-slt 33888 df-bday 33889 df-sslt 34017 df-scut 34019 df-made 34072 df-old 34073 df-left 34075 df-right 34076 |
This theorem is referenced by: noinds 34143 norecov 34145 noxpordpred 34151 no2indslem 34152 no3inds 34156 |
Copyright terms: Public domain | W3C validator |