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Mirrors > Home > MPE Home > Th. List > Mathboxes > lrrecval2 | Structured version Visualization version GIF version |
Description: Next, we establish an alternate expression for 𝑅. (Contributed by Scott Fenton, 19-Aug-2024.) |
Ref | Expression |
---|---|
lrrec.1 | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} |
Ref | Expression |
---|---|
lrrecval2 | ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝑅𝐵 ↔ ( bday ‘𝐴) ∈ ( bday ‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lrrec.1 | . . 3 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} | |
2 | 1 | lrrecval 33678 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝑅𝐵 ↔ 𝐴 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))) |
3 | lrold 33668 | . . . 4 ⊢ (𝐵 ∈ No → (( L ‘𝐵) ∪ ( R ‘𝐵)) = ( O ‘( bday ‘𝐵))) | |
4 | 3 | adantl 485 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( L ‘𝐵) ∪ ( R ‘𝐵)) = ( O ‘( bday ‘𝐵))) |
5 | 4 | eleq2d 2837 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)) ↔ 𝐴 ∈ ( O ‘( bday ‘𝐵)))) |
6 | bdayelon 33568 | . . . 4 ⊢ ( bday ‘𝐵) ∈ On | |
7 | oldbday 33672 | . . . 4 ⊢ ((( bday ‘𝐵) ∈ On ∧ 𝐴 ∈ No ) → (𝐴 ∈ ( O ‘( bday ‘𝐵)) ↔ ( bday ‘𝐴) ∈ ( bday ‘𝐵))) | |
8 | 6, 7 | mpan 689 | . . 3 ⊢ (𝐴 ∈ No → (𝐴 ∈ ( O ‘( bday ‘𝐵)) ↔ ( bday ‘𝐴) ∈ ( bday ‘𝐵))) |
9 | 8 | adantr 484 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ∈ ( O ‘( bday ‘𝐵)) ↔ ( bday ‘𝐴) ∈ ( bday ‘𝐵))) |
10 | 2, 5, 9 | 3bitrd 308 | 1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝑅𝐵 ↔ ( bday ‘𝐴) ∈ ( bday ‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∪ cun 3858 class class class wbr 5036 {copab 5098 Oncon0 6174 ‘cfv 6340 No csur 33440 bday cbday 33442 O cold 33621 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-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: lrrecpo 33680 lrrecfr 33682 |
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