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Theorem 2elresin 5346
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 5338 . . . . . . . 8  |-  ( ( F  Fn  A  /\  <.
x ,  y >.  e.  F )  ->  x  e.  A )
2 fnop 5338 . . . . . . . 8  |-  ( ( G  Fn  B  /\  <.
x ,  z >.  e.  G )  ->  x  e.  B )
31, 2anim12i 338 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  <. x ,  y
>.  e.  F )  /\  ( G  Fn  B  /\  <. x ,  z
>.  e.  G ) )  ->  ( x  e.  A  /\  x  e.  B ) )
43an4s 588 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  ->  ( x  e.  A  /\  x  e.  B ) )
5 elin 3333 . . . . . 6  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
64, 5sylibr 134 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  ->  x  e.  ( A  i^i  B ) )
7 vex 2755 . . . . . . . 8  |-  y  e. 
_V
87opres 4934 . . . . . . 7  |-  ( x  e.  ( A  i^i  B )  ->  ( <. x ,  y >.  e.  ( F  |`  ( A  i^i  B ) )  <->  <. x ,  y >.  e.  F
) )
9 vex 2755 . . . . . . . 8  |-  z  e. 
_V
109opres 4934 . . . . . . 7  |-  ( x  e.  ( A  i^i  B )  ->  ( <. x ,  z >.  e.  ( G  |`  ( A  i^i  B ) )  <->  <. x ,  z >.  e.  G
) )
118, 10anbi12d 473 . . . . . 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 158 . . . . 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 115 . . 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 50 . 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 4949 . . . 4  |-  ( F  |`  ( A  i^i  B
) )  C_  F
1716sseli 3166 . . 3  |-  ( <.
x ,  y >.  e.  ( F  |`  ( A  i^i  B ) )  ->  <. x ,  y
>.  e.  F )
18 resss 4949 . . . 4  |-  ( G  |`  ( A  i^i  B
) )  C_  G
1918sseli 3166 . . 3  |-  ( <.
x ,  z >.  e.  ( G  |`  ( A  i^i  B ) )  ->  <. x ,  z
>.  e.  G )
2017, 19anim12i 338 . 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 142 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 104    <-> wb 105    e. wcel 2160    i^i cin 3143   <.cop 3610    |` cres 4646    Fn wfn 5230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-dm 4654  df-res 4656  df-fun 5237  df-fn 5238
This theorem is referenced by: (None)
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