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Mirrors > Home > MPE Home > Th. List > Mathboxes > lrrecse | Structured version Visualization version GIF version |
Description: Next, we show that 𝑅 is set-like over No . (Contributed by Scott Fenton, 19-Aug-2024.) |
Ref | Expression |
---|---|
lrrec.1 | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} |
Ref | Expression |
---|---|
lrrecse | ⊢ 𝑅 Se No |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-se 5488 | . 2 ⊢ (𝑅 Se No ↔ ∀𝑎 ∈ No {𝑏 ∈ No ∣ 𝑏𝑅𝑎} ∈ V) | |
2 | lrrec.1 | . . . . . 6 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} | |
3 | 2 | lrrecval 33679 | . . . . 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 | . . . . 5 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = ({𝑏 ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∩ No ) | |
7 | abid2 2894 | . . . . . 6 ⊢ {𝑏 ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = (( L ‘𝑎) ∪ ( R ‘𝑎)) | |
8 | 7 | ineq1i 4115 | . . . . 5 ⊢ ({𝑏 ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∩ No ) = ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No ) |
9 | 6, 8 | eqtri 2781 | . . . 4 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No ) |
10 | fvex 6676 | . . . . . 6 ⊢ ( L ‘𝑎) ∈ V | |
11 | fvex 6676 | . . . . . 6 ⊢ ( R ‘𝑎) ∈ V | |
12 | 10, 11 | unex 7473 | . . . . 5 ⊢ (( L ‘𝑎) ∪ ( R ‘𝑎)) ∈ V |
13 | 12 | inex1 5191 | . . . 4 ⊢ ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No ) ∈ V |
14 | 9, 13 | eqeltri 2848 | . . 3 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∈ V |
15 | 5, 14 | eqeltrdi 2860 | . 2 ⊢ (𝑎 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝑎} ∈ V) |
16 | 1, 15 | mprgbir 3085 | 1 ⊢ 𝑅 Se No |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 {cab 2735 {crab 3074 Vcvv 3409 ∪ cun 3858 ∩ cin 3859 class class class wbr 5036 {copab 5098 Se wse 5485 ‘cfv 6340 No csur 33441 L cleft 33624 R cright 33625 |
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 ax-un 7465 |
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-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-se 5488 df-iota 6299 df-fv 6348 |
This theorem is referenced by: noinds 33685 norecfn 33686 norecov 33687 noxpordse 33692 no2indslem 33694 no3indslem 33698 |
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