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Theorem opelco3 30207
Description: Alternate way of saying that an ordered pair is in a composition. (Contributed by Scott Fenton, 6-May-2018.)
Assertion
Ref Expression
opelco3  |-  ( <. A ,  B >.  e.  ( C  o.  D
)  <->  B  e.  ( C " ( D " { A } ) ) )

Proof of Theorem opelco3
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-br 4427 . 2  |-  ( A ( C  o.  D
) B  <->  <. A ,  B >.  e.  ( C  o.  D ) )
2 relco 5353 . . . 4  |-  Rel  ( C  o.  D )
3 brrelex12 4892 . . . 4  |-  ( ( Rel  ( C  o.  D )  /\  A
( C  o.  D
) B )  -> 
( A  e.  _V  /\  B  e.  _V )
)
42, 3mpan 674 . . 3  |-  ( A ( C  o.  D
) B  ->  ( A  e.  _V  /\  B  e.  _V ) )
5 snprc 4066 . . . . . 6  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
6 noel 3771 . . . . . . 7  |-  -.  B  e.  (/)
7 imaeq2 5184 . . . . . . . . . 10  |-  ( { A }  =  (/)  ->  ( D " { A } )  =  ( D " (/) ) )
87imaeq2d 5188 . . . . . . . . 9  |-  ( { A }  =  (/)  ->  ( C " ( D " { A }
) )  =  ( C " ( D
" (/) ) ) )
9 ima0 5203 . . . . . . . . . . 11  |-  ( D
" (/) )  =  (/)
109imaeq2i 5186 . . . . . . . . . 10  |-  ( C
" ( D " (/) ) )  =  ( C " (/) )
11 ima0 5203 . . . . . . . . . 10  |-  ( C
" (/) )  =  (/)
1210, 11eqtri 2458 . . . . . . . . 9  |-  ( C
" ( D " (/) ) )  =  (/)
138, 12syl6eq 2486 . . . . . . . 8  |-  ( { A }  =  (/)  ->  ( C " ( D " { A }
) )  =  (/) )
1413eleq2d 2499 . . . . . . 7  |-  ( { A }  =  (/)  ->  ( B  e.  ( C " ( D
" { A }
) )  <->  B  e.  (/) ) )
156, 14mtbiri 304 . . . . . 6  |-  ( { A }  =  (/)  ->  -.  B  e.  ( C " ( D
" { A }
) ) )
165, 15sylbi 198 . . . . 5  |-  ( -.  A  e.  _V  ->  -.  B  e.  ( C
" ( D " { A } ) ) )
1716con4i 133 . . . 4  |-  ( B  e.  ( C "
( D " { A } ) )  ->  A  e.  _V )
18 elex 3096 . . . 4  |-  ( B  e.  ( C "
( D " { A } ) )  ->  B  e.  _V )
1917, 18jca 534 . . 3  |-  ( B  e.  ( C "
( D " { A } ) )  -> 
( A  e.  _V  /\  B  e.  _V )
)
20 df-rex 2788 . . . . 5  |-  ( E. z  e.  ( D
" { A }
) z C B  <->  E. z ( z  e.  ( D " { A } )  /\  z C B ) )
21 vex 3090 . . . . . . . . . 10  |-  z  e. 
_V
22 elimasng 5214 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  z  e.  _V )  ->  ( z  e.  ( D " { A } )  <->  <. A , 
z >.  e.  D ) )
2321, 22mpan2 675 . . . . . . . . 9  |-  ( A  e.  _V  ->  (
z  e.  ( D
" { A }
)  <->  <. A ,  z
>.  e.  D ) )
24 df-br 4427 . . . . . . . . 9  |-  ( A D z  <->  <. A , 
z >.  e.  D )
2523, 24syl6bbr 266 . . . . . . . 8  |-  ( A  e.  _V  ->  (
z  e.  ( D
" { A }
)  <->  A D z ) )
2625adantr 466 . . . . . . 7  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( z  e.  ( D " { A } )  <->  A D
z ) )
2726anbi1d 709 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( z  e.  ( D " { A } )  /\  z C B )  <->  ( A D z  /\  z C B ) ) )
2827exbidv 1761 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( E. z ( z  e.  ( D
" { A }
)  /\  z C B )  <->  E. z
( A D z  /\  z C B ) ) )
2920, 28syl5rbb 261 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( E. z ( A D z  /\  z C B )  <->  E. z  e.  ( D " { A } ) z C B ) )
30 brcog 5021 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A ( C  o.  D ) B  <->  E. z ( A D z  /\  z C B ) ) )
31 elimag 5192 . . . . 5  |-  ( B  e.  _V  ->  ( B  e.  ( C " ( D " { A } ) )  <->  E. z  e.  ( D " { A } ) z C B ) )
3231adantl 467 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( B  e.  ( C " ( D
" { A }
) )  <->  E. z  e.  ( D " { A } ) z C B ) )
3329, 30, 323bitr4d 288 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A ( C  o.  D ) B  <-> 
B  e.  ( C
" ( D " { A } ) ) ) )
344, 19, 33pm5.21nii 354 . 2  |-  ( A ( C  o.  D
) B  <->  B  e.  ( C " ( D
" { A }
) ) )
351, 34bitr3i 254 1  |-  ( <. A ,  B >.  e.  ( C  o.  D
)  <->  B  e.  ( C " ( D " { A } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1870   E.wrex 2783   _Vcvv 3087   (/)c0 3767   {csn 4002   <.cop 4008   class class class wbr 4426   "cima 4857    o. ccom 4858   Rel wrel 4859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867
This theorem is referenced by: (None)
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