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Theorem dmfco 5609
Description: Domains of a function composition. (Contributed by NM, 27-Jan-1997.)
Assertion
Ref Expression
dmfco  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( A  e.  dom  ( F  o.  G
)  <->  ( G `  A )  e.  dom  F ) )

Proof of Theorem dmfco
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldm2g 4891 . . . 4  |-  ( A  e.  dom  G  -> 
( A  e.  dom  ( F  o.  G
)  <->  E. y <. A , 
y >.  e.  ( F  o.  G ) ) )
2 vex 2804 . . . . . 6  |-  y  e. 
_V
3 opelco2g 4867 . . . . . 6  |-  ( ( A  e.  dom  G  /\  y  e.  _V )  ->  ( <. A , 
y >.  e.  ( F  o.  G )  <->  E. x
( <. A ,  x >.  e.  G  /\  <. x ,  y >.  e.  F
) ) )
42, 3mpan2 652 . . . . 5  |-  ( A  e.  dom  G  -> 
( <. A ,  y
>.  e.  ( F  o.  G )  <->  E. x
( <. A ,  x >.  e.  G  /\  <. x ,  y >.  e.  F
) ) )
54exbidv 1616 . . . 4  |-  ( A  e.  dom  G  -> 
( E. y <. A ,  y >.  e.  ( F  o.  G
)  <->  E. y E. x
( <. A ,  x >.  e.  G  /\  <. x ,  y >.  e.  F
) ) )
61, 5bitrd 244 . . 3  |-  ( A  e.  dom  G  -> 
( A  e.  dom  ( F  o.  G
)  <->  E. y E. x
( <. A ,  x >.  e.  G  /\  <. x ,  y >.  e.  F
) ) )
76adantl 452 . 2  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( A  e.  dom  ( F  o.  G
)  <->  E. y E. x
( <. A ,  x >.  e.  G  /\  <. x ,  y >.  e.  F
) ) )
8 fvex 5555 . . . 4  |-  ( G `
 A )  e. 
_V
98eldm2 4893 . . 3  |-  ( ( G `  A )  e.  dom  F  <->  E. y <. ( G `  A
) ,  y >.  e.  F )
10 opeq1 3812 . . . . . . 7  |-  ( x  =  ( G `  A )  ->  <. x ,  y >.  =  <. ( G `  A ) ,  y >. )
1110eleq1d 2362 . . . . . 6  |-  ( x  =  ( G `  A )  ->  ( <. x ,  y >.  e.  F  <->  <. ( G `  A ) ,  y
>.  e.  F ) )
128, 11ceqsexv 2836 . . . . 5  |-  ( E. x ( x  =  ( G `  A
)  /\  <. x ,  y >.  e.  F
)  <->  <. ( G `  A ) ,  y
>.  e.  F )
13 eqcom 2298 . . . . . . . 8  |-  ( x  =  ( G `  A )  <->  ( G `  A )  =  x )
14 funopfvb 5582 . . . . . . . 8  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( ( G `  A )  =  x  <->  <. A ,  x >.  e.  G ) )
1513, 14syl5bb 248 . . . . . . 7  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( x  =  ( G `  A )  <->  <. A ,  x >.  e.  G ) )
1615anbi1d 685 . . . . . 6  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( ( x  =  ( G `  A
)  /\  <. x ,  y >.  e.  F
)  <->  ( <. A ,  x >.  e.  G  /\  <.
x ,  y >.  e.  F ) ) )
1716exbidv 1616 . . . . 5  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( E. x ( x  =  ( G `
 A )  /\  <.
x ,  y >.  e.  F )  <->  E. x
( <. A ,  x >.  e.  G  /\  <. x ,  y >.  e.  F
) ) )
1812, 17syl5bbr 250 . . . 4  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( <. ( G `  A ) ,  y
>.  e.  F  <->  E. x
( <. A ,  x >.  e.  G  /\  <. x ,  y >.  e.  F
) ) )
1918exbidv 1616 . . 3  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( E. y <.
( G `  A
) ,  y >.  e.  F  <->  E. y E. x
( <. A ,  x >.  e.  G  /\  <. x ,  y >.  e.  F
) ) )
209, 19syl5bb 248 . 2  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( ( G `  A )  e.  dom  F  <->  E. y E. x (
<. A ,  x >.  e.  G  /\  <. x ,  y >.  e.  F
) ) )
217, 20bitr4d 247 1  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( A  e.  dom  ( F  o.  G
)  <->  ( G `  A )  e.  dom  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   dom cdm 4705    o. ccom 4709   Fun wfun 5265   ` cfv 5271
This theorem is referenced by:  funressnfv  28096  dmfcoafv  28143  afvco2  28144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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