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Theorem opelco3 29061
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 4448 . 2  |-  ( A ( C  o.  D
) B  <->  <. A ,  B >.  e.  ( C  o.  D ) )
2 relco 5505 . . . 4  |-  Rel  ( C  o.  D )
3 brrelex12 5037 . . . 4  |-  ( ( Rel  ( C  o.  D )  /\  A
( C  o.  D
) B )  -> 
( A  e.  _V  /\  B  e.  _V )
)
42, 3mpan 670 . . 3  |-  ( A ( C  o.  D
) B  ->  ( A  e.  _V  /\  B  e.  _V ) )
5 snprc 4091 . . . . . 6  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
6 noel 3789 . . . . . . 7  |-  -.  B  e.  (/)
7 imaeq2 5333 . . . . . . . . . 10  |-  ( { A }  =  (/)  ->  ( D " { A } )  =  ( D " (/) ) )
87imaeq2d 5337 . . . . . . . . 9  |-  ( { A }  =  (/)  ->  ( C " ( D " { A }
) )  =  ( C " ( D
" (/) ) ) )
9 ima0 5352 . . . . . . . . . . 11  |-  ( D
" (/) )  =  (/)
109imaeq2i 5335 . . . . . . . . . 10  |-  ( C
" ( D " (/) ) )  =  ( C " (/) )
11 ima0 5352 . . . . . . . . . 10  |-  ( C
" (/) )  =  (/)
1210, 11eqtri 2496 . . . . . . . . 9  |-  ( C
" ( D " (/) ) )  =  (/)
138, 12syl6eq 2524 . . . . . . . 8  |-  ( { A }  =  (/)  ->  ( C " ( D " { A }
) )  =  (/) )
1413eleq2d 2537 . . . . . . 7  |-  ( { A }  =  (/)  ->  ( B  e.  ( C " ( D
" { A }
) )  <->  B  e.  (/) ) )
156, 14mtbiri 303 . . . . . 6  |-  ( { A }  =  (/)  ->  -.  B  e.  ( C " ( D
" { A }
) ) )
165, 15sylbi 195 . . . . 5  |-  ( -.  A  e.  _V  ->  -.  B  e.  ( C
" ( D " { A } ) ) )
1716con4i 130 . . . 4  |-  ( B  e.  ( C "
( D " { A } ) )  ->  A  e.  _V )
18 elex 3122 . . . 4  |-  ( B  e.  ( C "
( D " { A } ) )  ->  B  e.  _V )
1917, 18jca 532 . . 3  |-  ( B  e.  ( C "
( D " { A } ) )  -> 
( A  e.  _V  /\  B  e.  _V )
)
20 df-rex 2820 . . . . 5  |-  ( E. z  e.  ( D
" { A }
) z C B  <->  E. z ( z  e.  ( D " { A } )  /\  z C B ) )
21 vex 3116 . . . . . . . . . 10  |-  z  e. 
_V
22 elimasng 5363 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  z  e.  _V )  ->  ( z  e.  ( D " { A } )  <->  <. A , 
z >.  e.  D ) )
2321, 22mpan2 671 . . . . . . . . 9  |-  ( A  e.  _V  ->  (
z  e.  ( D
" { A }
)  <->  <. A ,  z
>.  e.  D ) )
24 df-br 4448 . . . . . . . . 9  |-  ( A D z  <->  <. A , 
z >.  e.  D )
2523, 24syl6bbr 263 . . . . . . . 8  |-  ( A  e.  _V  ->  (
z  e.  ( D
" { A }
)  <->  A D z ) )
2625adantr 465 . . . . . . 7  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( z  e.  ( D " { A } )  <->  A D
z ) )
2726anbi1d 704 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( z  e.  ( D " { A } )  /\  z C B )  <->  ( A D z  /\  z C B ) ) )
2827exbidv 1690 . . . . 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 258 . . . 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 5169 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A ( C  o.  D ) B  <->  E. z ( A D z  /\  z C B ) ) )
31 elimag 5341 . . . . 5  |-  ( B  e.  _V  ->  ( B  e.  ( C " ( D " { A } ) )  <->  E. z  e.  ( D " { A } ) z C B ) )
3231adantl 466 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( B  e.  ( C " ( D
" { A }
) )  <->  E. z  e.  ( D " { A } ) z C B ) )
3329, 30, 323bitr4d 285 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A ( C  o.  D ) B  <-> 
B  e.  ( C
" ( D " { A } ) ) ) )
344, 19, 33pm5.21nii 353 . 2  |-  ( A ( C  o.  D
) B  <->  B  e.  ( C " ( D
" { A }
) ) )
351, 34bitr3i 251 1  |-  ( <. A ,  B >.  e.  ( C  o.  D
)  <->  B  e.  ( C " ( D " { A } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   E.wrex 2815   _Vcvv 3113   (/)c0 3785   {csn 4027   <.cop 4033   class class class wbr 4447   "cima 5002    o. ccom 5003   Rel wrel 5004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012
This theorem is referenced by: (None)
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