![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5633 | . 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 27423 | . . . . 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 3440 | . . 3 ⊢ (𝑎 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝑎} = {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))}) |
6 | dfrab2 4311 | . . . . 5 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = ({𝑏 ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∩ No ) | |
7 | abid2 2872 | . . . . . 6 ⊢ {𝑏 ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = (( L ‘𝑎) ∪ ( R ‘𝑎)) | |
8 | 7 | ineq1i 4209 | . . . . 5 ⊢ ({𝑏 ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∩ No ) = ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No ) |
9 | 6, 8 | eqtri 2761 | . . . 4 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No ) |
10 | fvex 6905 | . . . . . 6 ⊢ ( L ‘𝑎) ∈ V | |
11 | fvex 6905 | . . . . . 6 ⊢ ( R ‘𝑎) ∈ V | |
12 | 10, 11 | unex 7733 | . . . . 5 ⊢ (( L ‘𝑎) ∪ ( R ‘𝑎)) ∈ V |
13 | 12 | inex1 5318 | . . . 4 ⊢ ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No ) ∈ V |
14 | 9, 13 | eqeltri 2830 | . . 3 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∈ V |
15 | 5, 14 | eqeltrdi 2842 | . 2 ⊢ (𝑎 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝑎} ∈ V) |
16 | 1, 15 | mprgbir 3069 | 1 ⊢ 𝑅 Se No |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∈ wcel 2107 {cab 2710 {crab 3433 Vcvv 3475 ∪ cun 3947 ∩ cin 3948 class class class wbr 5149 {copab 5211 Se wse 5630 ‘cfv 6544 No csur 27143 L cleft 27340 R cright 27341 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
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 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-se 5633 df-iota 6496 df-fv 6552 |
This theorem is referenced by: noinds 27429 norecfn 27430 norecov 27431 noxpordse 27436 no2indslem 27438 no3inds 27442 |
Copyright terms: Public domain | W3C validator |