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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 32859 | . . 3 ⊢ LB𝑅 = UB◡𝑅 | |
2 | 1 | breqi 4929 | . 2 ⊢ (𝑆LB𝑅𝐴 ↔ 𝑆UB◡𝑅𝐴) |
3 | brub.1 | . . 3 ⊢ 𝑆 ∈ V | |
4 | brub.2 | . . 3 ⊢ 𝐴 ∈ V | |
5 | 3, 4 | brub 32936 | . 2 ⊢ (𝑆UB◡𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥◡𝑅𝐴) |
6 | vex 3412 | . . . 4 ⊢ 𝑥 ∈ V | |
7 | 6, 4 | brcnv 5597 | . . 3 ⊢ (𝑥◡𝑅𝐴 ↔ 𝐴𝑅𝑥) |
8 | 7 | ralbii 3109 | . 2 ⊢ (∀𝑥 ∈ 𝑆 𝑥◡𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥) |
9 | 2, 5, 8 | 3bitri 289 | 1 ⊢ (𝑆LB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∈ wcel 2050 ∀wral 3082 Vcvv 3409 class class class wbr 4923 ◡ccnv 5400 UBcub 32834 LBclb 32835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5054 ax-nul 5061 ax-pr 5180 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-br 4924 df-opab 4986 df-eprel 5311 df-xp 5407 df-cnv 5409 df-co 5410 df-ub 32858 df-lb 32859 |
This theorem is referenced by: (None) |
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