Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > brlb | Structured version Visualization version GIF version |
Description: Binary relation form of the lower bound functor. (Contributed by Scott Fenton, 3-May-2018.) |
Ref | Expression |
---|---|
brub.1 | ⊢ 𝑆 ∈ V |
brub.2 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
brlb | ⊢ (𝑆LB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-lb 33865 | . . 3 ⊢ LB𝑅 = UB◡𝑅 | |
2 | 1 | breqi 5045 | . 2 ⊢ (𝑆LB𝑅𝐴 ↔ 𝑆UB◡𝑅𝐴) |
3 | brub.1 | . . 3 ⊢ 𝑆 ∈ V | |
4 | brub.2 | . . 3 ⊢ 𝐴 ∈ V | |
5 | 3, 4 | brub 33942 | . 2 ⊢ (𝑆UB◡𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥◡𝑅𝐴) |
6 | vex 3402 | . . . 4 ⊢ 𝑥 ∈ V | |
7 | 6, 4 | brcnv 5736 | . . 3 ⊢ (𝑥◡𝑅𝐴 ↔ 𝐴𝑅𝑥) |
8 | 7 | ralbii 3078 | . 2 ⊢ (∀𝑥 ∈ 𝑆 𝑥◡𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥) |
9 | 2, 5, 8 | 3bitri 300 | 1 ⊢ (𝑆LB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∈ wcel 2112 ∀wral 3051 Vcvv 3398 class class class wbr 5039 ◡ccnv 5535 UBcub 33840 LBclb 33841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-eprel 5445 df-xp 5542 df-cnv 5544 df-co 5545 df-ub 33864 df-lb 33865 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |