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Theorem dmfco 5798
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 5067 . . . 4  |-  ( A  e.  dom  G  -> 
( A  e.  dom  ( F  o.  G
)  <->  E. y <. A , 
y >.  e.  ( F  o.  G ) ) )
2 vex 2960 . . . . . 6  |-  y  e. 
_V
3 opelco2g 5041 . . . . . 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 654 . . . . 5  |-  ( A  e.  dom  G  -> 
( <. A ,  y
>.  e.  ( F  o.  G )  <->  E. x
( <. A ,  x >.  e.  G  /\  <. x ,  y >.  e.  F
) ) )
54exbidv 1637 . . . 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 246 . . 3  |-  ( A  e.  dom  G  -> 
( A  e.  dom  ( F  o.  G
)  <->  E. y E. x
( <. A ,  x >.  e.  G  /\  <. x ,  y >.  e.  F
) ) )
76adantl 454 . 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 5743 . . . 4  |-  ( G `
 A )  e. 
_V
98eldm2 5069 . . 3  |-  ( ( G `  A )  e.  dom  F  <->  E. y <. ( G `  A
) ,  y >.  e.  F )
10 opeq1 3985 . . . . . . 7  |-  ( x  =  ( G `  A )  ->  <. x ,  y >.  =  <. ( G `  A ) ,  y >. )
1110eleq1d 2503 . . . . . 6  |-  ( x  =  ( G `  A )  ->  ( <. x ,  y >.  e.  F  <->  <. ( G `  A ) ,  y
>.  e.  F ) )
128, 11ceqsexv 2992 . . . . 5  |-  ( E. x ( x  =  ( G `  A
)  /\  <. x ,  y >.  e.  F
)  <->  <. ( G `  A ) ,  y
>.  e.  F )
13 eqcom 2439 . . . . . . . 8  |-  ( x  =  ( G `  A )  <->  ( G `  A )  =  x )
14 funopfvb 5771 . . . . . . . 8  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( ( G `  A )  =  x  <->  <. A ,  x >.  e.  G ) )
1513, 14syl5bb 250 . . . . . . 7  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( x  =  ( G `  A )  <->  <. A ,  x >.  e.  G ) )
1615anbi1d 687 . . . . . 6  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( ( x  =  ( G `  A
)  /\  <. x ,  y >.  e.  F
)  <->  ( <. A ,  x >.  e.  G  /\  <.
x ,  y >.  e.  F ) ) )
1716exbidv 1637 . . . . 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 252 . . . 4  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( <. ( G `  A ) ,  y
>.  e.  F  <->  E. x
( <. A ,  x >.  e.  G  /\  <. x ,  y >.  e.  F
) ) )
1918exbidv 1637 . . 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 250 . 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 249 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 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   _Vcvv 2957   <.cop 3818   dom cdm 4879    o. ccom 4883   Fun wfun 5449   ` cfv 5455
This theorem is referenced by:  funressnfv  27969  dmfcoafv  28016  afvco2  28017
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fn 5458  df-fv 5463
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