Theorem brabg 2813

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 2812	. 2 |- ((A e. C /\ B e. D) -> (<.A, B>. e. {<.x, y>. | ph} <-> ch))
4		df-br 2615	. . 3 |- (ARB <-> <.A, B>. e. R)
5		brabg.5	. . . 4 |- R = {<.x, y>. | ph}
6	5	eleq2i 1535	. . 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 531	1 |- ((A e. C /\ B e. D) -> (ARB <-> ch))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  <.cop 2407   class class class wbr 2614  {copab 2661

This theorem is referenced by:  brab 2816  ideqg 3271  f1owe 3896  breng 4363  brdomg 4364  ltprord 5114  clim 6923  lmbr 7880  hlim2 8999  cmbrt 9467  leopg 9993  cvbrt 10147  mdbrt 10159  dmdbrt 10164  hmph 10447

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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