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Theorem brabgaf 28219
Description: The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.) (Revised by Thierry Arnoux, 17-May-2020.)
Hypotheses
Ref Expression
brabgaf.0  |-  F/ x ps
brabgaf.1  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
brabgaf.2  |-  R  =  { <. x ,  y
>.  |  ph }
Assertion
Ref Expression
brabgaf  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A R B  <->  ps ) )
Distinct variable groups:    x, y, A    x, B, y    ps, y
Allowed substitution hints:    ph( x, y)    ps( x)    R( x, y)    V( x, y)    W( x, y)

Proof of Theorem brabgaf
StepHypRef Expression
1 df-br 4424 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  R )
2 brabgaf.2 . . . 4  |-  R  =  { <. x ,  y
>.  |  ph }
32eleq2i 2499 . . 3  |-  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } )
41, 3bitri 252 . 2  |-  ( A R B  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } )
5 elopab 4728 . . 3  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) )
6 elisset 3091 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
7 elisset 3091 . . . 4  |-  ( B  e.  W  ->  E. y 
y  =  B )
8 eeanv 2047 . . . . 5  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  <->  ( E. x  x  =  A  /\  E. y 
y  =  B ) )
9 nfe1 1894 . . . . . . 7  |-  F/ x E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  ph )
10 brabgaf.0 . . . . . . 7  |-  F/ x ps
119, 10nfbi 1994 . . . . . 6  |-  F/ x
( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps )
12 nfe1 1894 . . . . . . . . 9  |-  F/ y E. y ( <. A ,  B >.  = 
<. x ,  y >.  /\  ph )
1312nfex 2008 . . . . . . . 8  |-  F/ y E. x E. y
( <. A ,  B >.  =  <. x ,  y
>.  /\  ph )
14 nfv 1755 . . . . . . . 8  |-  F/ y ps
1513, 14nfbi 1994 . . . . . . 7  |-  F/ y ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps )
16 opeq12 4189 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B )  -> 
<. x ,  y >.  =  <. A ,  B >. )
17 copsexg 4706 . . . . . . . . . 10  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  ( ph  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) ) )
1817eqcoms 2434 . . . . . . . . 9  |-  ( <.
x ,  y >.  =  <. A ,  B >.  ->  ( ph  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) ) )
1916, 18syl 17 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) ) )
20 brabgaf.1 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
2119, 20bitr3d 258 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps ) )
2215, 21exlimi 1972 . . . . . 6  |-  ( E. y ( x  =  A  /\  y  =  B )  ->  ( E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  ph )  <->  ps )
)
2311, 22exlimi 1972 . . . . 5  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  ->  ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps ) )
248, 23sylbir 216 . . . 4  |-  ( ( E. x  x  =  A  /\  E. y 
y  =  B )  ->  ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps ) )
256, 7, 24syl2an 479 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps ) )
265, 25syl5bb 260 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<->  ps ) )
274, 26syl5bb 260 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A R B  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1657   F/wnf 1661    e. wcel 1872   <.cop 4004   class class class wbr 4423   {copab 4481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424  df-opab 4483
This theorem is referenced by:  fmptcof2  28262
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