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Theorem opelco3 29382
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 4457 . 2  |-  ( A ( C  o.  D
) B  <->  <. A ,  B >.  e.  ( C  o.  D ) )
2 relco 5511 . . . 4  |-  Rel  ( C  o.  D )
3 brrelex12 5046 . . . 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 4095 . . . . . 6  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
6 noel 3797 . . . . . . 7  |-  -.  B  e.  (/)
7 imaeq2 5343 . . . . . . . . . 10  |-  ( { A }  =  (/)  ->  ( D " { A } )  =  ( D " (/) ) )
87imaeq2d 5347 . . . . . . . . 9  |-  ( { A }  =  (/)  ->  ( C " ( D " { A }
) )  =  ( C " ( D
" (/) ) ) )
9 ima0 5362 . . . . . . . . . . 11  |-  ( D
" (/) )  =  (/)
109imaeq2i 5345 . . . . . . . . . 10  |-  ( C
" ( D " (/) ) )  =  ( C " (/) )
11 ima0 5362 . . . . . . . . . 10  |-  ( C
" (/) )  =  (/)
1210, 11eqtri 2486 . . . . . . . . 9  |-  ( C
" ( D " (/) ) )  =  (/)
138, 12syl6eq 2514 . . . . . . . 8  |-  ( { A }  =  (/)  ->  ( C " ( D " { A }
) )  =  (/) )
1413eleq2d 2527 . . . . . . 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 3118 . . . 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 2813 . . . . 5  |-  ( E. z  e.  ( D
" { A }
) z C B  <->  E. z ( z  e.  ( D " { A } )  /\  z C B ) )
21 vex 3112 . . . . . . . . . 10  |-  z  e. 
_V
22 elimasng 5373 . . . . . . . . . 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 4457 . . . . . . . . 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 1715 . . . . 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 5179 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A ( C  o.  D ) B  <->  E. z ( A D z  /\  z C B ) ) )
31 elimag 5351 . . . . 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 1395   E.wex 1613    e. wcel 1819   E.wrex 2808   _Vcvv 3109   (/)c0 3793   {csn 4032   <.cop 4038   class class class wbr 4456   "cima 5011    o. ccom 5012   Rel wrel 5013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021
This theorem is referenced by: (None)
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