![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > brub | Structured version Visualization version GIF version |
Description: Binary relation form of the upper bound functor. (Contributed by Scott Fenton, 3-May-2018.) |
Ref | Expression |
---|---|
brub.1 | ⊢ 𝑆 ∈ V |
brub.2 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
brub | ⊢ (𝑆UB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brub.1 | . . . . 5 ⊢ 𝑆 ∈ V | |
2 | brub.2 | . . . . 5 ⊢ 𝐴 ∈ V | |
3 | brxp 5682 | . . . . 5 ⊢ (𝑆(V × V)𝐴 ↔ (𝑆 ∈ V ∧ 𝐴 ∈ V)) | |
4 | 1, 2, 3 | mpbir2an 710 | . . . 4 ⊢ 𝑆(V × V)𝐴 |
5 | brdif 5159 | . . . 4 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))𝐴 ↔ (𝑆(V × V)𝐴 ∧ ¬ 𝑆((V ∖ 𝑅) ∘ ◡ E )𝐴)) | |
6 | 4, 5 | mpbiran 708 | . . 3 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))𝐴 ↔ ¬ 𝑆((V ∖ 𝑅) ∘ ◡ E )𝐴) |
7 | 1, 2 | coepr 34382 | . . 3 ⊢ (𝑆((V ∖ 𝑅) ∘ ◡ E )𝐴 ↔ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴) |
8 | 6, 7 | xchbinx 334 | . 2 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))𝐴 ↔ ¬ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴) |
9 | df-ub 34507 | . . 3 ⊢ UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E )) | |
10 | 9 | breqi 5112 | . 2 ⊢ (𝑆UB𝑅𝐴 ↔ 𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))𝐴) |
11 | brv 5430 | . . . . . 6 ⊢ 𝑥V𝐴 | |
12 | brdif 5159 | . . . . . 6 ⊢ (𝑥(V ∖ 𝑅)𝐴 ↔ (𝑥V𝐴 ∧ ¬ 𝑥𝑅𝐴)) | |
13 | 11, 12 | mpbiran 708 | . . . . 5 ⊢ (𝑥(V ∖ 𝑅)𝐴 ↔ ¬ 𝑥𝑅𝐴) |
14 | 13 | rexbii 3094 | . . . 4 ⊢ (∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴 ↔ ∃𝑥 ∈ 𝑆 ¬ 𝑥𝑅𝐴) |
15 | rexnal 3100 | . . . 4 ⊢ (∃𝑥 ∈ 𝑆 ¬ 𝑥𝑅𝐴 ↔ ¬ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴) | |
16 | 14, 15 | bitri 275 | . . 3 ⊢ (∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴 ↔ ¬ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴) |
17 | 16 | con2bii 358 | . 2 ⊢ (∀𝑥 ∈ 𝑆 𝑥𝑅𝐴 ↔ ¬ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴) |
18 | 8, 10, 17 | 3bitr4i 303 | 1 ⊢ (𝑆UB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∈ wcel 2107 ∀wral 3061 ∃wrex 3070 Vcvv 3444 ∖ cdif 3908 class class class wbr 5106 E cep 5537 × cxp 5632 ◡ccnv 5633 ∘ ccom 5638 UBcub 34483 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-eprel 5538 df-xp 5640 df-cnv 5642 df-co 5643 df-ub 34507 |
This theorem is referenced by: brlb 34586 |
Copyright terms: Public domain | W3C validator |