Theorem brab 2821

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 2818	. 2 |- ((A e. V /\ B e. V) -> (ARB <-> ch))
7	1, 2, 6	mp2an 697	1 |- (ARB <-> ch)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  Vcvv 1811   class class class wbr 2619  {copab 2666

This theorem is referenced by:  epelc 2833  opbrop 3238  f1oweALT 3906  aceq3 4733  zornlem 4795  brdom7disj 4804  brdom6disj 4805  ltresr 5258  hlim 9056

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667

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