Theorem brab 2118

Description: The law of concretion for a binary relation.

Hypotheses

Ref	Expression
opelopab.1	⊢ A ∈ V
opelopab.2	⊢ B ∈ V
opelopab.3	⊢ (x = A → (φ ↔ ψ))
opelopab.4	⊢ (y = B → (ψ ↔ χ))
brab.5	⊢ R = {⟨x, y⟩∣φ}

Assertion

Ref	Expression
brab	⊢ (ARB ↔ χ)

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

Proof of Theorem brab

Step	Hyp	Ref	Expression
1		opelopab.1	. 2 ⊢ A ∈ V
2		opelopab.2	. 2 ⊢ B ∈ V
3		opelopab.3	. . 3 ⊢ (x = A → (φ ↔ ψ))
4		opelopab.4	. . 3 ⊢ (y = B → (ψ ↔ χ))
5		brab.5	. . 3 ⊢ R = {⟨x, y⟩∣φ}
6	3, 4, 5	brabg 2116	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (ARB ↔ χ))
7	1, 2, 6	mp2an 520	1 ⊢ (ARB ↔ χ)

Syntax hints:   → wi 2   ↔ wb 127   = wceq 1091   ∈ wcel 1092  Vcvv 1348   class class class wbr 2054  {copab 2055

This theorem is referenced by:  epelc 2123  opbrop 2472  f1oweOLD 2944  zorn2lem 3610  ltresr 4052  clim 4877  hlim 5108

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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