Theorem brabga 5472

Description: The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
opelopabga.1	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
brabga.2	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Assertion

Ref	Expression
brabga	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴𝑅𝐵 ↔ 𝜓))

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

Step	Hyp	Ref	Expression
1		df-br 5090	. . 3 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		brabga.2	. . . 4 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
3	2	eleq2i 2823	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
4	1, 3	bitri 275	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5		opelopabga.1	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
6	5	opelopabga 5471	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓))
7	4, 6	bitrid 283	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴𝑅𝐵 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ⟨cop 4579   class class class wbr 5089  {copab 5151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152

This theorem is referenced by:  braba  5475  brabg  5477  epelg  5515  brcog  5805  fmptco  7062  ofrfvalg  7618  isfsupp  9249  wemaplem1  9432  oemapval  9573  wemapwe  9587  fpwwe2lem2  10523  fpwwelem  10536  clim  15401  rlim  15402  vdwmc  16890  isstruct2  17060  brssc  17721  isfunc  17771  isfull  17819  isfth  17823  ipole  18440  eqgval  19089  frgpuplem  19684  dvdsr  20280  islindf  21749  ulmval  26316  hpgbr  28738  isausgr  29142  issubgr  29249  isrgr  29538  isrusgr  29540  istrlson  29683  upgrwlkdvspth  29717  ispthson  29720  isspthson  29721  erclwwlkeq  29998  erclwwlkneq  30047  hlimi  31168  isinftm  33150  brfldext  33658  brfinext  33665  finextfldext  33677  bralgext  33710  fldext2chn  33741  constrextdg2lem  33761  metidv  33905  ismntoplly  34038  brae  34254  braew  34255  brfae  34261  satfbrsuc  35410  prv  35472  bj-epelg  37112  bj-ideqgALT  37202  bj-idreseq  37206  bj-idreseqb  37207  bj-ideqg1ALT  37209  brsucmap  38478  brcoss  38532  brcoels  38536  brdmqss  38742  aks6d1c1p1  42199  climf  45721  climf2  45763  nelbr  47373  iscllaw  48288  iscomlaw  48289  isasslaw  48291  islininds  48546  lindepsnlininds  48552