Theorem brab 2816

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 2813	. 2 |- ((A e. V /\ B e. V) -> (ARB <-> ch))
7	1, 2, 6	mp2an 696	1 |- (ARB <-> ch)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 954   e. wcel 956  Vcvv 1807   class class class wbr 2614  {copab 2661

This theorem is referenced by:  epelc 2828  opbrop 3233  f1oweALT 3897  aceq3 4713  zornlem 4775  brdom7disj 4784  brdom6disj 4785  ltresr 5238  hlim 8995

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662

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