Theorem brab 2827

Description: The law of concretion for a binary relation.

Hypotheses

Ref	Expression
opelopab.1	|- A e. V
opelopab.2	|- B e. V
opelopab.3	|- (x = A -> (ph <-> ps))
opelopab.4	|- (y = B -> (ps <-> ch))
brab.5	|- R = {<.x, y>. | ph}

Assertion

Ref	Expression
brab	|- (ARB <-> ch)

Distinct variable groups:   x,y,A   x,B,y   ch,x,y

Proof of Theorem brab

Step	Hyp	Ref	Expression
1		opelopab.1	. 2 |- A e. V
2		opelopab.2	. 2 |- B e. V
3		opelopab.3	. . 3 |- (x = A -> (ph <-> ps))
4		opelopab.4	. . 3 |- (y = B -> (ps <-> ch))
5		brab.5	. . 3 |- R = {<.x, y>. | ph}
6	3, 4, 5	brabg 2824	. 2 |- ((A e. V /\ B e. V) -> (ARB <-> ch))
7	1, 2, 6	mp2an 699	1 |- (ARB <-> ch)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960  Vcvv 1814   class class class wbr 2624  {copab 2671

This theorem is referenced by:  epelc 2839  opbrop 3244  f1oweALT 3912  aceq3 4743  zornlem 4805  brdom7disj 4814  brdom6disj 4815  ltresr 5270  hlim 9051  inposetlem 10475

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672

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