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Theorem brub 34585
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 5682 . . . . 5 (𝑆(V × V)𝐴 ↔ (𝑆 ∈ V ∧ 𝐴 ∈ V))
41, 2, 3mpbir2an 710 . . . 4 𝑆(V × V)𝐴
5 brdif 5159 . . . 4 (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ E ))𝐴 ↔ (𝑆(V × V)𝐴 ∧ ¬ 𝑆((V ∖ 𝑅) ∘ E )𝐴))
64, 5mpbiran 708 . . 3 (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ E ))𝐴 ↔ ¬ 𝑆((V ∖ 𝑅) ∘ E )𝐴)
71, 2coepr 34382 . . 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 34507 . . 3 UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ E ))
109breqi 5112 . 2 (𝑆UB𝑅𝐴𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ E ))𝐴)
11 brv 5430 . . . . . 6 𝑥V𝐴
12 brdif 5159 . . . . . 6 (𝑥(V ∖ 𝑅)𝐴 ↔ (𝑥V𝐴 ∧ ¬ 𝑥𝑅𝐴))
1311, 12mpbiran 708 . . . . 5 (𝑥(V ∖ 𝑅)𝐴 ↔ ¬ 𝑥𝑅𝐴)
1413rexbii 3094 . . . 4 (∃𝑥𝑆 𝑥(V ∖ 𝑅)𝐴 ↔ ∃𝑥𝑆 ¬ 𝑥𝑅𝐴)
15 rexnal 3100 . . . 4 (∃𝑥𝑆 ¬ 𝑥𝑅𝐴 ↔ ¬ ∀𝑥𝑆 𝑥𝑅𝐴)
1614, 15bitri 275 . . 3 (∃𝑥𝑆 𝑥(V ∖ 𝑅)𝐴 ↔ ¬ ∀𝑥𝑆 𝑥𝑅𝐴)
1716con2bii 358 . 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 205  wcel 2107  wral 3061  wrex 3070  Vcvv 3444  cdif 3908   class class class wbr 5106   E cep 5537   × cxp 5632  ccnv 5633  ccom 5638  UBcub 34483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-eprel 5538  df-xp 5640  df-cnv 5642  df-co 5643  df-ub 34507
This theorem is referenced by:  brlb  34586
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