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Theorem brub 35422
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 5716 . . . . 5 (𝑆(V × V)𝐴 ↔ (𝑆 ∈ V ∧ 𝐴 ∈ V))
41, 2, 3mpbir2an 708 . . . 4 𝑆(V × V)𝐴
5 brdif 5192 . . . 4 (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ E ))𝐴 ↔ (𝑆(V × V)𝐴 ∧ ¬ 𝑆((V ∖ 𝑅) ∘ E )𝐴))
64, 5mpbiran 706 . . 3 (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ E ))𝐴 ↔ ¬ 𝑆((V ∖ 𝑅) ∘ E )𝐴)
71, 2coepr 35219 . . 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 35344 . . 3 UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ E ))
109breqi 5145 . 2 (𝑆UB𝑅𝐴𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ E ))𝐴)
11 brv 5463 . . . . . 6 𝑥V𝐴
12 brdif 5192 . . . . . 6 (𝑥(V ∖ 𝑅)𝐴 ↔ (𝑥V𝐴 ∧ ¬ 𝑥𝑅𝐴))
1311, 12mpbiran 706 . . . . 5 (𝑥(V ∖ 𝑅)𝐴 ↔ ¬ 𝑥𝑅𝐴)
1413rexbii 3086 . . . 4 (∃𝑥𝑆 𝑥(V ∖ 𝑅)𝐴 ↔ ∃𝑥𝑆 ¬ 𝑥𝑅𝐴)
15 rexnal 3092 . . . 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 205  wcel 2098  wral 3053  wrex 3062  Vcvv 3466  cdif 3938   class class class wbr 5139   E cep 5570   × cxp 5665  ccnv 5666  ccom 5671  UBcub 35320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-eprel 5571  df-xp 5673  df-cnv 5675  df-co 5676  df-ub 35344
This theorem is referenced by:  brlb  35423
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