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Mirrors > Home > MPE Home > Th. List > 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 5590 | . 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 27254 | . . . . 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 3415 | . . 3 ⊢ (𝑎 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝑎} = {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))}) |
6 | dfrab2 4271 | . . . . 5 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = ({𝑏 ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∩ No ) | |
7 | abid2 2876 | . . . . . 6 ⊢ {𝑏 ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = (( L ‘𝑎) ∪ ( R ‘𝑎)) | |
8 | 7 | ineq1i 4169 | . . . . 5 ⊢ ({𝑏 ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∩ No ) = ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No ) |
9 | 6, 8 | eqtri 2765 | . . . 4 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No ) |
10 | fvex 6856 | . . . . . 6 ⊢ ( L ‘𝑎) ∈ V | |
11 | fvex 6856 | . . . . . 6 ⊢ ( R ‘𝑎) ∈ V | |
12 | 10, 11 | unex 7681 | . . . . 5 ⊢ (( L ‘𝑎) ∪ ( R ‘𝑎)) ∈ V |
13 | 12 | inex1 5275 | . . . 4 ⊢ ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No ) ∈ V |
14 | 9, 13 | eqeltri 2834 | . . 3 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∈ V |
15 | 5, 14 | eqeltrdi 2846 | . 2 ⊢ (𝑎 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝑎} ∈ V) |
16 | 1, 15 | mprgbir 3072 | 1 ⊢ 𝑅 Se No |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∈ wcel 2107 {cab 2714 {crab 3408 Vcvv 3446 ∪ cun 3909 ∩ cin 3910 class class class wbr 5106 {copab 5168 Se wse 5587 ‘cfv 6497 No csur 26991 L cleft 27178 R cright 27179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-se 5590 df-iota 6449 df-fv 6505 |
This theorem is referenced by: noinds 27260 norecfn 27261 norecov 27262 noxpordse 27267 no2indslem 27269 no3inds 27273 |
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