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Theorem brub 35955
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 5734 . . . . 5 (𝑆(V × V)𝐴 ↔ (𝑆 ∈ V ∧ 𝐴 ∈ V))
41, 2, 3mpbir2an 711 . . . 4 𝑆(V × V)𝐴
5 brdif 5196 . . . 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 35753 . . 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 35877 . . 3 UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ E ))
109breqi 5149 . 2 (𝑆UB𝑅𝐴𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ E ))𝐴)
11 brv 5477 . . . . . 6 𝑥V𝐴
12 brdif 5196 . . . . . 6 (𝑥(V ∖ 𝑅)𝐴 ↔ (𝑥V𝐴 ∧ ¬ 𝑥𝑅𝐴))
1311, 12mpbiran 709 . . . . 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 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 2108  wral 3061  wrex 3070  Vcvv 3480  cdif 3948   class class class wbr 5143   E cep 5583   × cxp 5683  ccnv 5684  ccom 5689  UBcub 35853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-eprel 5584  df-xp 5691  df-cnv 5693  df-co 5694  df-ub 35877
This theorem is referenced by:  brlb  35956
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