Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > noxpordpred | Structured version Visualization version GIF version |
Description: Next we calculate the predecessor class of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.) |
Ref | Expression |
---|---|
noxpord.1 | ⊢ 𝑅 = {〈𝑎, 𝑏〉 ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} |
noxpord.2 | ⊢ 𝑆 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} |
Ref | Expression |
---|---|
noxpordpred | ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → Pred(𝑆, ( No × No ), 〈𝐴, 𝐵〉) = ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noxpord.2 | . . 3 ⊢ 𝑆 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} | |
2 | 1 | xpord2pred 33359 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → Pred(𝑆, ( No × No ), 〈𝐴, 𝐵〉) = (((Pred(𝑅, No , 𝐴) ∪ {𝐴}) × (Pred(𝑅, No , 𝐵) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) |
3 | noxpord.1 | . . . . . . 7 ⊢ 𝑅 = {〈𝑎, 𝑏〉 ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} | |
4 | 3 | lrrecpred 33683 | . . . . . 6 ⊢ (𝐴 ∈ No → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴))) |
5 | 4 | adantr 484 | . . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴))) |
6 | 5 | uneq1d 4069 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (Pred(𝑅, No , 𝐴) ∪ {𝐴}) = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴})) |
7 | 3 | lrrecpred 33683 | . . . . . 6 ⊢ (𝐵 ∈ No → Pred(𝑅, No , 𝐵) = (( L ‘𝐵) ∪ ( R ‘𝐵))) |
8 | 7 | adantl 485 | . . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → Pred(𝑅, No , 𝐵) = (( L ‘𝐵) ∪ ( R ‘𝐵))) |
9 | 8 | uneq1d 4069 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (Pred(𝑅, No , 𝐵) ∪ {𝐵}) = ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) |
10 | 6, 9 | xpeq12d 5559 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((Pred(𝑅, No , 𝐴) ∪ {𝐴}) × (Pred(𝑅, No , 𝐵) ∪ {𝐵})) = (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))) |
11 | 10 | difeq1d 4029 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (((Pred(𝑅, No , 𝐴) ∪ {𝐴}) × (Pred(𝑅, No , 𝐵) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}) = ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) |
12 | 2, 11 | eqtrd 2793 | 1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → Pred(𝑆, ( No × No ), 〈𝐴, 𝐵〉) = ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2951 ∖ cdif 3857 ∪ cun 3858 {csn 4525 〈cop 4531 class class class wbr 5036 {copab 5098 × cxp 5526 Predcpred 6130 ‘cfv 6340 1st c1st 7697 2nd c2nd 7698 No csur 33440 L cleft 33623 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-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 |
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 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7699 df-2nd 7700 df-wrecs 7963 df-recs 8024 df-1o 8118 df-2o 8119 df-no 33443 df-slt 33444 df-bday 33445 df-sslt 33573 df-scut 33575 df-made 33625 df-old 33626 df-left 33628 df-right 33629 |
This theorem is referenced by: norec2ov 33696 |
Copyright terms: Public domain | W3C validator |