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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 5716 | . . . . 5 ⊢ (𝑆(V × V)𝐴 ↔ (𝑆 ∈ V ∧ 𝐴 ∈ V)) | |
4 | 1, 2, 3 | mpbir2an 708 | . . . 4 ⊢ 𝑆(V × V)𝐴 |
5 | brdif 5192 | . . . 4 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))𝐴 ↔ (𝑆(V × V)𝐴 ∧ ¬ 𝑆((V ∖ 𝑅) ∘ ◡ E )𝐴)) | |
6 | 4, 5 | mpbiran 706 | . . 3 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))𝐴 ↔ ¬ 𝑆((V ∖ 𝑅) ∘ ◡ E )𝐴) |
7 | 1, 2 | coepr 35219 | . . 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 35344 | . . 3 ⊢ UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E )) | |
10 | 9 | breqi 5145 | . 2 ⊢ (𝑆UB𝑅𝐴 ↔ 𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))𝐴) |
11 | brv 5463 | . . . . . 6 ⊢ 𝑥V𝐴 | |
12 | brdif 5192 | . . . . . 6 ⊢ (𝑥(V ∖ 𝑅)𝐴 ↔ (𝑥V𝐴 ∧ ¬ 𝑥𝑅𝐴)) | |
13 | 11, 12 | mpbiran 706 | . . . . 5 ⊢ (𝑥(V ∖ 𝑅)𝐴 ↔ ¬ 𝑥𝑅𝐴) |
14 | 13 | rexbii 3086 | . . . 4 ⊢ (∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴 ↔ ∃𝑥 ∈ 𝑆 ¬ 𝑥𝑅𝐴) |
15 | rexnal 3092 | . . . 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 205 ∈ wcel 2098 ∀wral 3053 ∃wrex 3062 Vcvv 3466 ∖ cdif 3938 class class class wbr 5139 E cep 5570 × cxp 5665 ◡ccnv 5666 ∘ ccom 5671 UBcub 35320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-eprel 5571 df-xp 5673 df-cnv 5675 df-co 5676 df-ub 35344 |
This theorem is referenced by: brlb 35423 |
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