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Theorem brub 35936
Description: Binary relation form of the upper bound functor. (Contributed by Scott Fenton, 3-May-2018.)
Hypotheses
Ref Expression
brub.1 𝑆 ∈ V
brub.2 𝐴 ∈ V
Assertion
Ref Expression
brub (𝑆UB𝑅𝐴 ↔ ∀𝑥𝑆 𝑥𝑅𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅   𝑥,𝑆

Proof of Theorem brub
StepHypRef Expression
1 brub.1 . . . . 5 𝑆 ∈ V
2 brub.2 . . . . 5 𝐴 ∈ V
3 brxp 5738 . . . . 5 (𝑆(V × V)𝐴 ↔ (𝑆 ∈ V ∧ 𝐴 ∈ V))
41, 2, 3mpbir2an 711 . . . 4 𝑆(V × V)𝐴
5 brdif 5201 . . . 4 (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ E ))𝐴 ↔ (𝑆(V × V)𝐴 ∧ ¬ 𝑆((V ∖ 𝑅) ∘ E )𝐴))
64, 5mpbiran 709 . . 3 (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ E ))𝐴 ↔ ¬ 𝑆((V ∖ 𝑅) ∘ E )𝐴)
71, 2coepr 35733 . . 3 (𝑆((V ∖ 𝑅) ∘ E )𝐴 ↔ ∃𝑥𝑆 𝑥(V ∖ 𝑅)𝐴)
86, 7xchbinx 334 . 2 (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ E ))𝐴 ↔ ¬ ∃𝑥𝑆 𝑥(V ∖ 𝑅)𝐴)
9 df-ub 35858 . . 3 UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ E ))
109breqi 5154 . 2 (𝑆UB𝑅𝐴𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ E ))𝐴)
11 brv 5483 . . . . . 6 𝑥V𝐴
12 brdif 5201 . . . . . 6 (𝑥(V ∖ 𝑅)𝐴 ↔ (𝑥V𝐴 ∧ ¬ 𝑥𝑅𝐴))
1311, 12mpbiran 709 . . . . 5 (𝑥(V ∖ 𝑅)𝐴 ↔ ¬ 𝑥𝑅𝐴)
1413rexbii 3092 . . . 4 (∃𝑥𝑆 𝑥(V ∖ 𝑅)𝐴 ↔ ∃𝑥𝑆 ¬ 𝑥𝑅𝐴)
15 rexnal 3098 . . . 4 (∃𝑥𝑆 ¬ 𝑥𝑅𝐴 ↔ ¬ ∀𝑥𝑆 𝑥𝑅𝐴)
1614, 15bitri 275 . . 3 (∃𝑥𝑆 𝑥(V ∖ 𝑅)𝐴 ↔ ¬ ∀𝑥𝑆 𝑥𝑅𝐴)
1716con2bii 357 . 2 (∀𝑥𝑆 𝑥𝑅𝐴 ↔ ¬ ∃𝑥𝑆 𝑥(V ∖ 𝑅)𝐴)
188, 10, 173bitr4i 303 1 (𝑆UB𝑅𝐴 ↔ ∀𝑥𝑆 𝑥𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  cdif 3960   class class class wbr 5148   E cep 5588   × cxp 5687  ccnv 5688  ccom 5693  UBcub 35834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-eprel 5589  df-xp 5695  df-cnv 5697  df-co 5698  df-ub 35858
This theorem is referenced by:  brlb  35937
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