| 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 5734 | . . . . 5 ⊢ (𝑆(V × V)𝐴 ↔ (𝑆 ∈ V ∧ 𝐴 ∈ V)) | |
| 4 | 1, 2, 3 | mpbir2an 711 | . . . 4 ⊢ 𝑆(V × V)𝐴 |
| 5 | brdif 5196 | . . . 4 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))𝐴 ↔ (𝑆(V × V)𝐴 ∧ ¬ 𝑆((V ∖ 𝑅) ∘ ◡ E )𝐴)) | |
| 6 | 4, 5 | mpbiran 709 | . . 3 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))𝐴 ↔ ¬ 𝑆((V ∖ 𝑅) ∘ ◡ E )𝐴) |
| 7 | 1, 2 | coepr 35753 | . . 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 35877 | . . 3 ⊢ UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E )) | |
| 10 | 9 | breqi 5149 | . 2 ⊢ (𝑆UB𝑅𝐴 ↔ 𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))𝐴) |
| 11 | brv 5477 | . . . . . 6 ⊢ 𝑥V𝐴 | |
| 12 | brdif 5196 | . . . . . 6 ⊢ (𝑥(V ∖ 𝑅)𝐴 ↔ (𝑥V𝐴 ∧ ¬ 𝑥𝑅𝐴)) | |
| 13 | 11, 12 | mpbiran 709 | . . . . 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 357 | . 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 206 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ∖ cdif 3948 class class class wbr 5143 E cep 5583 × cxp 5683 ◡ccnv 5684 ∘ ccom 5689 UBcub 35853 |
| 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 |
| This theorem is referenced by: brlb 35956 |
| Copyright terms: Public domain | W3C validator |