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Mirrors > Home > MPE Home > Th. List > Mathboxes > brub | Structured version Visualization version GIF version |
Description: Binary relationship 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 5071 | . . . . 5 ⊢ (𝑆(V × V)𝐴 ↔ (𝑆 ∈ V ∧ 𝐴 ∈ V)) | |
4 | 1, 2, 3 | mpbir2an 957 | . . . 4 ⊢ 𝑆(V × V)𝐴 |
5 | brdif 4635 | . . . 4 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))𝐴 ↔ (𝑆(V × V)𝐴 ∧ ¬ 𝑆((V ∖ 𝑅) ∘ ◡ E )𝐴)) | |
6 | 4, 5 | mpbiran 955 | . . 3 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))𝐴 ↔ ¬ 𝑆((V ∖ 𝑅) ∘ ◡ E )𝐴) |
7 | 1, 2 | coepr 30895 | . . 3 ⊢ (𝑆((V ∖ 𝑅) ∘ ◡ E )𝐴 ↔ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴) |
8 | 6, 7 | xchbinx 323 | . 2 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))𝐴 ↔ ¬ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴) |
9 | df-ub 31152 | . . 3 ⊢ UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E )) | |
10 | 9 | breqi 4589 | . 2 ⊢ (𝑆UB𝑅𝐴 ↔ 𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))𝐴) |
11 | brv 31154 | . . . . . 6 ⊢ 𝑥V𝐴 | |
12 | brdif 4635 | . . . . . 6 ⊢ (𝑥(V ∖ 𝑅)𝐴 ↔ (𝑥V𝐴 ∧ ¬ 𝑥𝑅𝐴)) | |
13 | 11, 12 | mpbiran 955 | . . . . 5 ⊢ (𝑥(V ∖ 𝑅)𝐴 ↔ ¬ 𝑥𝑅𝐴) |
14 | 13 | rexbii 3023 | . . . 4 ⊢ (∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴 ↔ ∃𝑥 ∈ 𝑆 ¬ 𝑥𝑅𝐴) |
15 | rexnal 2978 | . . . 4 ⊢ (∃𝑥 ∈ 𝑆 ¬ 𝑥𝑅𝐴 ↔ ¬ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴) | |
16 | 14, 15 | bitri 263 | . . 3 ⊢ (∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴 ↔ ¬ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴) |
17 | 16 | con2bii 346 | . 2 ⊢ (∀𝑥 ∈ 𝑆 𝑥𝑅𝐴 ↔ ¬ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴) |
18 | 8, 10, 17 | 3bitr4i 291 | 1 ⊢ (𝑆UB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∖ cdif 3537 class class class wbr 4583 E cep 4947 × cxp 5036 ◡ccnv 5037 ∘ ccom 5042 UBcub 31128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-eprel 4949 df-xp 5044 df-cnv 5046 df-co 5047 df-ub 31152 |
This theorem is referenced by: brlb 31232 |
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