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Theorem 2elresin 5081
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 5073 . . . . . . . 8  |-  ( ( F  Fn  A  /\  <.
x ,  y >.  e.  F )  ->  x  e.  A )
2 fnop 5073 . . . . . . . 8  |-  ( ( G  Fn  B  /\  <.
x ,  z >.  e.  G )  ->  x  e.  B )
31, 2anim12i 331 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  <. x ,  y
>.  e.  F )  /\  ( G  Fn  B  /\  <. x ,  z
>.  e.  G ) )  ->  ( x  e.  A  /\  x  e.  B ) )
43an4s 553 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  ->  ( x  e.  A  /\  x  e.  B ) )
5 elin 3169 . . . . . 6  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
64, 5sylibr 132 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  ->  x  e.  ( A  i^i  B ) )
7 vex 2617 . . . . . . . 8  |-  y  e. 
_V
87opres 4683 . . . . . . 7  |-  ( x  e.  ( A  i^i  B )  ->  ( <. x ,  y >.  e.  ( F  |`  ( A  i^i  B ) )  <->  <. x ,  y >.  e.  F
) )
9 vex 2617 . . . . . . . 8  |-  z  e. 
_V
109opres 4683 . . . . . . 7  |-  ( x  e.  ( A  i^i  B )  ->  ( <. x ,  z >.  e.  ( G  |`  ( A  i^i  B ) )  <->  <. x ,  z >.  e.  G
) )
118, 10anbi12d 457 . . . . . 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 156 . . . . 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 14 . . . 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 113 . . 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 49 . 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 4697 . . . 4  |-  ( F  |`  ( A  i^i  B
) )  C_  F
1716sseli 3008 . . 3  |-  ( <.
x ,  y >.  e.  ( F  |`  ( A  i^i  B ) )  ->  <. x ,  y
>.  e.  F )
18 resss 4697 . . . 4  |-  ( G  |`  ( A  i^i  B
) )  C_  G
1918sseli 3008 . . 3  |-  ( <.
x ,  z >.  e.  ( G  |`  ( A  i^i  B ) )  ->  <. x ,  z
>.  e.  G )
2017, 19anim12i 331 . 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 140 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    /\ wa 102    <-> wb 103    e. wcel 1436    i^i cin 2985   <.cop 3428    |` cres 4406    Fn wfn 4967
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-br 3815  df-opab 3869  df-xp 4410  df-rel 4411  df-dm 4414  df-res 4416  df-fun 4974  df-fn 4975
This theorem is referenced by: (None)
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