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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 35468 | . . 3 ⊢ LB𝑅 = UB◡𝑅 | |
2 | 1 | breqi 5149 | . 2 ⊢ (𝑆LB𝑅𝐴 ↔ 𝑆UB◡𝑅𝐴) |
3 | brub.1 | . . 3 ⊢ 𝑆 ∈ V | |
4 | brub.2 | . . 3 ⊢ 𝐴 ∈ V | |
5 | 3, 4 | brub 35545 | . 2 ⊢ (𝑆UB◡𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥◡𝑅𝐴) |
6 | vex 3474 | . . . 4 ⊢ 𝑥 ∈ V | |
7 | 6, 4 | brcnv 5880 | . . 3 ⊢ (𝑥◡𝑅𝐴 ↔ 𝐴𝑅𝑥) |
8 | 7 | ralbii 3089 | . 2 ⊢ (∀𝑥 ∈ 𝑆 𝑥◡𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥) |
9 | 2, 5, 8 | 3bitri 297 | 1 ⊢ (𝑆LB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2099 ∀wral 3057 Vcvv 3470 class class class wbr 5143 ◡ccnv 5672 UBcub 35443 LBclb 35444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-br 5144 df-opab 5206 df-eprel 5577 df-xp 5679 df-cnv 5681 df-co 5682 df-ub 35467 df-lb 35468 |
This theorem is referenced by: (None) |
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