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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 35878 | . . 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 35955 | . 2 ⊢ (𝑆UB◡𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥◡𝑅𝐴) |
| 6 | vex 3484 | . . . 4 ⊢ 𝑥 ∈ V | |
| 7 | 6, 4 | brcnv 5893 | . . 3 ⊢ (𝑥◡𝑅𝐴 ↔ 𝐴𝑅𝑥) |
| 8 | 7 | ralbii 3093 | . 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 206 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 class class class wbr 5143 ◡ccnv 5684 UBcub 35853 LBclb 35854 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-eprel 5584 df-xp 5691 df-cnv 5693 df-co 5694 df-ub 35877 df-lb 35878 |
| This theorem is referenced by: (None) |
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