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Mirrors > Home > MPE Home > Th. List > Mathboxes > brlb | Structured version Visualization version GIF version |
Description: Binary relationship 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 31153 | . . 3 ⊢ LB𝑅 = UB◡𝑅 | |
2 | 1 | breqi 4589 | . 2 ⊢ (𝑆LB𝑅𝐴 ↔ 𝑆UB◡𝑅𝐴) |
3 | brub.1 | . . 3 ⊢ 𝑆 ∈ V | |
4 | brub.2 | . . 3 ⊢ 𝐴 ∈ V | |
5 | 3, 4 | brub 31231 | . 2 ⊢ (𝑆UB◡𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥◡𝑅𝐴) |
6 | vex 3176 | . . . 4 ⊢ 𝑥 ∈ V | |
7 | 6, 4 | brcnv 5227 | . . 3 ⊢ (𝑥◡𝑅𝐴 ↔ 𝐴𝑅𝑥) |
8 | 7 | ralbii 2963 | . 2 ⊢ (∀𝑥 ∈ 𝑆 𝑥◡𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥) |
9 | 2, 5, 8 | 3bitri 285 | 1 ⊢ (𝑆LB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 class class class wbr 4583 ◡ccnv 5037 UBcub 31128 LBclb 31129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-eprel 4949 df-xp 5044 df-cnv 5046 df-co 5047 df-ub 31152 df-lb 31153 |
This theorem is referenced by: (None) |
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