Theorem brabg 2116

Description: The law of concretion for a binary relation.

Hypotheses

Ref	Expression
opelopabg.1	⊢ (x = A → (φ ↔ ψ))
opelopabg.2	⊢ (y = B → (ψ ↔ χ))
brabg.5	⊢ R = {⟨x, y⟩∣φ}

Assertion

Ref	Expression
brabg	⊢ ((A ∈ C ∧ B ∈ D) → (ARB ↔ χ))

Distinct variable group(s):   x,y,A   x,B,y   χ,x,y

Proof of Theorem brabg

Step	Hyp	Ref	Expression
1		opelopabg.1	. . 3 ⊢ (x = A → (φ ↔ ψ))
2		opelopabg.2	. . 3 ⊢ (y = B → (ψ ↔ χ))
3	1, 2	opelopabg 2115	. 2 ⊢ ((A ∈ C ∧ B ∈ D) → (⟨A, B⟩ ∈ {⟨x, y⟩∣φ} ↔ χ))
4		df-br 2063	. . 3 ⊢ (ARB ↔ ⟨A, B⟩ ∈ R)
5		brabg.5	. . . 4 ⊢ R = {⟨x, y⟩∣φ}
6	5	eleq2i 1153	. . 3 ⊢ (⟨A, B⟩ ∈ R ↔ ⟨A, B⟩ ∈ {⟨x, y⟩∣φ})
7	4, 6	bitr 151	. 2 ⊢ (ARB ↔ ⟨A, B⟩ ∈ {⟨x, y⟩∣φ})
8	3, 7	syl5bb 410	1 ⊢ ((A ∈ C ∧ B ∈ D) → (ARB ↔ χ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ⟨cop 1810   class class class wbr 2054  {copab 2055

This theorem is referenced by:  brab 2118  ideqg 2126  f1owe 2943  breng 3280  brdomg 3281  ltprord 3928  clim2 4881  hlim2 5112  cmbrt 5494  cvbrt 5714  mdbr 5726

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098

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