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Theorem brub 33528
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 5565 . . . . 5 (𝑆(V × V)𝐴 ↔ (𝑆 ∈ V ∧ 𝐴 ∈ V))
41, 2, 3mpbir2an 710 . . . 4 𝑆(V × V)𝐴
5 brdif 5083 . . . 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 33101 . . 3 (𝑆((V ∖ 𝑅) ∘ E )𝐴 ↔ ∃𝑥𝑆 𝑥(V ∖ 𝑅)𝐴)
86, 7xchbinx 337 . 2 (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ E ))𝐴 ↔ ¬ ∃𝑥𝑆 𝑥(V ∖ 𝑅)𝐴)
9 df-ub 33450 . . 3 UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ E ))
109breqi 5036 . 2 (𝑆UB𝑅𝐴𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ E ))𝐴)
11 brv 5329 . . . . . 6 𝑥V𝐴
12 brdif 5083 . . . . . 6 (𝑥(V ∖ 𝑅)𝐴 ↔ (𝑥V𝐴 ∧ ¬ 𝑥𝑅𝐴))
1311, 12mpbiran 708 . . . . 5 (𝑥(V ∖ 𝑅)𝐴 ↔ ¬ 𝑥𝑅𝐴)
1413rexbii 3210 . . . 4 (∃𝑥𝑆 𝑥(V ∖ 𝑅)𝐴 ↔ ∃𝑥𝑆 ¬ 𝑥𝑅𝐴)
15 rexnal 3201 . . . 4 (∃𝑥𝑆 ¬ 𝑥𝑅𝐴 ↔ ¬ ∀𝑥𝑆 𝑥𝑅𝐴)
1614, 15bitri 278 . . 3 (∃𝑥𝑆 𝑥(V ∖ 𝑅)𝐴 ↔ ¬ ∀𝑥𝑆 𝑥𝑅𝐴)
1716con2bii 361 . 2 (∀𝑥𝑆 𝑥𝑅𝐴 ↔ ¬ ∃𝑥𝑆 𝑥(V ∖ 𝑅)𝐴)
188, 10, 173bitr4i 306 1 (𝑆UB𝑅𝐴 ↔ ∀𝑥𝑆 𝑥𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wcel 2111  wral 3106  wrex 3107  Vcvv 3441  cdif 3878   class class class wbr 5030   E cep 5429   × cxp 5517  ccnv 5518  ccom 5523  UBcub 33426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-eprel 5430  df-xp 5525  df-cnv 5527  df-co 5528  df-ub 33450
This theorem is referenced by:  brlb  33529
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