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Theorem 2elresin 5355
Description: Membership in two functions restricted by each other's domain. (Contributed by NM, 8-Aug-1994.)
Assertion
Ref Expression
2elresin  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G )  <->  ( <. x ,  y >.  e.  ( F  |`  ( A  i^i  B ) )  /\  <.
x ,  z >.  e.  ( G  |`  ( A  i^i  B ) ) ) ) )

Proof of Theorem 2elresin
StepHypRef Expression
1 fnop 5347 . . . . . . . 8  |-  ( ( F  Fn  A  /\  <.
x ,  y >.  e.  F )  ->  x  e.  A )
2 fnop 5347 . . . . . . . 8  |-  ( ( G  Fn  B  /\  <.
x ,  z >.  e.  G )  ->  x  e.  B )
31, 2anim12i 549 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  <. x ,  y
>.  e.  F )  /\  ( G  Fn  B  /\  <. x ,  z
>.  e.  G ) )  ->  ( x  e.  A  /\  x  e.  B ) )
43an4s 799 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  ->  ( x  e.  A  /\  x  e.  B ) )
5 elin 3358 . . . . . 6  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
64, 5sylibr 203 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  ->  x  e.  ( A  i^i  B ) )
7 vex 2791 . . . . . . . 8  |-  y  e. 
_V
87opres 4964 . . . . . . 7  |-  ( x  e.  ( A  i^i  B )  ->  ( <. x ,  y >.  e.  ( F  |`  ( A  i^i  B ) )  <->  <. x ,  y >.  e.  F
) )
9 vex 2791 . . . . . . . 8  |-  z  e. 
_V
109opres 4964 . . . . . . 7  |-  ( x  e.  ( A  i^i  B )  ->  ( <. x ,  z >.  e.  ( G  |`  ( A  i^i  B ) )  <->  <. x ,  z >.  e.  G
) )
118, 10anbi12d 691 . . . . . 6  |-  ( x  e.  ( A  i^i  B )  ->  ( ( <. x ,  y >.  e.  ( F  |`  ( A  i^i  B ) )  /\  <. x ,  z
>.  e.  ( G  |`  ( A  i^i  B ) ) )  <->  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) ) )
1211biimprd 214 . . . . 5  |-  ( x  e.  ( A  i^i  B )  ->  ( ( <. x ,  y >.  e.  F  /\  <. x ,  z >.  e.  G
)  ->  ( <. x ,  y >.  e.  ( F  |`  ( A  i^i  B ) )  /\  <.
x ,  z >.  e.  ( G  |`  ( A  i^i  B ) ) ) ) )
136, 12syl 15 . . . 4  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  ->  ( ( <.
x ,  y >.  e.  F  /\  <. x ,  z >.  e.  G
)  ->  ( <. x ,  y >.  e.  ( F  |`  ( A  i^i  B ) )  /\  <.
x ,  z >.  e.  ( G  |`  ( A  i^i  B ) ) ) ) )
1413ex 423 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G )  -> 
( ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G )  -> 
( <. x ,  y
>.  e.  ( F  |`  ( A  i^i  B ) )  /\  <. x ,  z >.  e.  ( G  |`  ( A  i^i  B ) ) ) ) ) )
1514pm2.43d 44 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G )  -> 
( <. x ,  y
>.  e.  ( F  |`  ( A  i^i  B ) )  /\  <. x ,  z >.  e.  ( G  |`  ( A  i^i  B ) ) ) ) )
16 resss 4979 . . . 4  |-  ( F  |`  ( A  i^i  B
) )  C_  F
1716sseli 3176 . . 3  |-  ( <.
x ,  y >.  e.  ( F  |`  ( A  i^i  B ) )  ->  <. x ,  y
>.  e.  F )
18 resss 4979 . . . 4  |-  ( G  |`  ( A  i^i  B
) )  C_  G
1918sseli 3176 . . 3  |-  ( <.
x ,  z >.  e.  ( G  |`  ( A  i^i  B ) )  ->  <. x ,  z
>.  e.  G )
2017, 19anim12i 549 . 2  |-  ( (
<. x ,  y >.  e.  ( F  |`  ( A  i^i  B ) )  /\  <. x ,  z
>.  e.  ( G  |`  ( A  i^i  B ) ) )  ->  ( <. x ,  y >.  e.  F  /\  <. x ,  z >.  e.  G
) )
2115, 20impbid1 194 1  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G )  <->  ( <. x ,  y >.  e.  ( F  |`  ( A  i^i  B ) )  /\  <.
x ,  z >.  e.  ( G  |`  ( A  i^i  B ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684    i^i cin 3151   <.cop 3643    |` cres 4691    Fn wfn 5250
This theorem is referenced by:  tfrlem5  6396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-dm 4699  df-res 4701  df-fun 5257  df-fn 5258
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