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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 5603 | . . . . 5 ⊢ (𝑆(V × V)𝐴 ↔ (𝑆 ∈ V ∧ 𝐴 ∈ V)) | |
4 | 1, 2, 3 | mpbir2an 709 | . . . 4 ⊢ 𝑆(V × V)𝐴 |
5 | brdif 5121 | . . . 4 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))𝐴 ↔ (𝑆(V × V)𝐴 ∧ ¬ 𝑆((V ∖ 𝑅) ∘ ◡ E )𝐴)) | |
6 | 4, 5 | mpbiran 707 | . . 3 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))𝐴 ↔ ¬ 𝑆((V ∖ 𝑅) ∘ ◡ E )𝐴) |
7 | 1, 2 | coepr 32990 | . . 3 ⊢ (𝑆((V ∖ 𝑅) ∘ ◡ E )𝐴 ↔ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴) |
8 | 6, 7 | xchbinx 336 | . 2 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))𝐴 ↔ ¬ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴) |
9 | df-ub 33339 | . . 3 ⊢ UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E )) | |
10 | 9 | breqi 5074 | . 2 ⊢ (𝑆UB𝑅𝐴 ↔ 𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))𝐴) |
11 | brv 5366 | . . . . . 6 ⊢ 𝑥V𝐴 | |
12 | brdif 5121 | . . . . . 6 ⊢ (𝑥(V ∖ 𝑅)𝐴 ↔ (𝑥V𝐴 ∧ ¬ 𝑥𝑅𝐴)) | |
13 | 11, 12 | mpbiran 707 | . . . . 5 ⊢ (𝑥(V ∖ 𝑅)𝐴 ↔ ¬ 𝑥𝑅𝐴) |
14 | 13 | rexbii 3249 | . . . 4 ⊢ (∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴 ↔ ∃𝑥 ∈ 𝑆 ¬ 𝑥𝑅𝐴) |
15 | rexnal 3240 | . . . 4 ⊢ (∃𝑥 ∈ 𝑆 ¬ 𝑥𝑅𝐴 ↔ ¬ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴) | |
16 | 14, 15 | bitri 277 | . . 3 ⊢ (∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴 ↔ ¬ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴) |
17 | 16 | con2bii 360 | . 2 ⊢ (∀𝑥 ∈ 𝑆 𝑥𝑅𝐴 ↔ ¬ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴) |
18 | 8, 10, 17 | 3bitr4i 305 | 1 ⊢ (𝑆UB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 Vcvv 3496 ∖ cdif 3935 class class class wbr 5068 E cep 5466 × cxp 5555 ◡ccnv 5556 ∘ ccom 5561 UBcub 33315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-eprel 5467 df-xp 5563 df-cnv 5565 df-co 5566 df-ub 33339 |
This theorem is referenced by: brlb 33418 |
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