Theorem brabga 5497

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 5111	. . 3 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		brabga.2	. . . 4 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
3	2	eleq2i 2821	. . 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 5496	. 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 1540   ∈ wcel 2109  ⟨cop 4598   class class class wbr 5110  {copab 5172

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173

This theorem is referenced by:  braba  5500  brabg  5502  epelg  5542  brcog  5833  fmptco  7104  ofrfvalg  7664  isfsupp  9323  wemaplem1  9506  oemapval  9643  wemapwe  9657  fpwwe2lem2  10592  fpwwelem  10605  clim  15467  rlim  15468  vdwmc  16956  isstruct2  17126  brssc  17783  isfunc  17833  isfull  17881  isfth  17885  ipole  18500  eqgval  19116  frgpuplem  19709  dvdsr  20278  islindf  21728  ulmval  26296  hpgbr  28694  isausgr  29098  issubgr  29205  isrgr  29494  isrusgr  29496  istrlson  29642  upgrwlkdvspth  29676  ispthson  29679  isspthson  29680  erclwwlkeq  29954  erclwwlkneq  30003  hlimi  31124  isinftm  33142  brfldext  33648  brfinext  33655  fldext2chn  33725  constrextdg2lem  33745  metidv  33889  ismntoplly  34022  brae  34238  braew  34239  brfae  34245  satfbrsuc  35360  prv  35422  bj-epelg  37063  bj-ideqgALT  37153  bj-idreseq  37157  bj-idreseqb  37158  bj-ideqg1ALT  37160  brcoss  38429  brcoels  38433  brdmqss  38644  aks6d1c1p1  42102  climf  45627  climf2  45671  nelbr  47279  iscllaw  48181  iscomlaw  48182  isasslaw  48184  islininds  48439  lindepsnlininds  48445