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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 33235 | . . 3 ⊢ LB𝑅 = UB◡𝑅 | |
2 | 1 | breqi 5063 | . 2 ⊢ (𝑆LB𝑅𝐴 ↔ 𝑆UB◡𝑅𝐴) |
3 | brub.1 | . . 3 ⊢ 𝑆 ∈ V | |
4 | brub.2 | . . 3 ⊢ 𝐴 ∈ V | |
5 | 3, 4 | brub 33312 | . 2 ⊢ (𝑆UB◡𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥◡𝑅𝐴) |
6 | vex 3495 | . . . 4 ⊢ 𝑥 ∈ V | |
7 | 6, 4 | brcnv 5746 | . . 3 ⊢ (𝑥◡𝑅𝐴 ↔ 𝐴𝑅𝑥) |
8 | 7 | ralbii 3162 | . 2 ⊢ (∀𝑥 ∈ 𝑆 𝑥◡𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥) |
9 | 2, 5, 8 | 3bitri 298 | 1 ⊢ (𝑆LB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 class class class wbr 5057 ◡ccnv 5547 UBcub 33210 LBclb 33211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-eprel 5458 df-xp 5554 df-cnv 5556 df-co 5557 df-ub 33234 df-lb 33235 |
This theorem is referenced by: (None) |
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