Theorem brabg 2818

Description: The law of concretion for a binary relation.

Hypotheses

Ref	Expression
opelopabg.1	|- (x = A -> (ph <-> ps))
opelopabg.2	|- (y = B -> (ps <-> ch))
brabg.5	|- R = {<.x, y>. | ph}

Assertion

Ref	Expression
brabg	|- ((A e. C /\ B e. D) -> (ARB <-> ch))

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

Proof of Theorem brabg

Step	Hyp	Ref	Expression
1		opelopabg.1	. . 3 |- (x = A -> (ph <-> ps))
2		opelopabg.2	. . 3 |- (y = B -> (ps <-> ch))
3	1, 2	opelopabg 2817	. 2 |- ((A e. C /\ B e. D) -> (<.A, B>. e. {<.x, y>. | ph} <-> ch))
4		df-br 2620	. . 3 |- (ARB <-> <.A, B>. e. R)
5		brabg.5	. . . 4 |- R = {<.x, y>. | ph}
6	5	eleq2i 1538	. . 3 |- (<.A, B>. e. R <-> <.A, B>. e. {<.x, y>. | ph})
7	4, 6	bitr 173	. 2 |- (ARB <-> <.A, B>. e. {<.x, y>. | ph})
8	3, 7	syl5bb 532	1 |- ((A e. C /\ B e. D) -> (ARB <-> ch))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  <.cop 2411   class class class wbr 2619  {copab 2666

This theorem is referenced by:  brab 2821  ideqg 3276  f1owe 3905  breng 4375  brdomg 4376  ltprord 5134  clim 6977  lmbr 7928  hlim2 9060  cmbrt 9527  leopg 10055  cvbrt 10209  mdbrt 10221  dmdbrt 10226  hmph 10524

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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