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Theorem brabga 5122
Description: The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
opelopabga.1 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
brabga.2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Assertion
Ref Expression
brabga ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem brabga
StepHypRef Expression
1 df-br 4787 . . 3 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 brabga.2 . . . 4 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
32eleq2i 2842 . . 3 (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
41, 3bitri 264 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5 opelopabga.1 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
65opelopabga 5121 . 2 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓))
74, 6syl5bb 272 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  cop 4322   class class class wbr 4786  {copab 4846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847
This theorem is referenced by:  braba  5125  brabg  5127  epelg  5163  brcog  5427  fmptco  6539  ofrfval  7052  isfsupp  8435  wemaplem1  8607  oemapval  8744  wemapwe  8758  fpwwe2lem2  9656  fpwwelem  9669  clim  14433  rlim  14434  vdwmc  15889  isstruct2  16074  brssc  16681  isfunc  16731  isfull  16777  isfth  16781  ipole  17366  eqgval  17851  frgpuplem  18392  dvdsr  18854  islindf  20368  ulmval  24354  hpgbr  25873  isausgr  26281  issubgr  26386  isrgr  26690  isrusgr  26692  istrlson  26838  upgrwlkdvspth  26870  ispthson  26873  isspthson  26874  erclwwlkeq  27168  erclwwlkneq  27225  hlimi  28385  isinftm  30075  metidv  30275  ismntoplly  30409  brae  30644  braew  30645  brfae  30651  brcoss  34528  brcoels  34532  climf  40372  climf2  40416  nelbr  41814  iscllaw  42353  iscomlaw  42354  isasslaw  42356  islininds  42763  lindepsnlininds  42769
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