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Theorem 2elresin 5190
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 5182 . . . . . . . 8  |-  ( ( F  Fn  A  /\  <.
x ,  y >.  e.  F )  ->  x  e.  A )
2 fnop 5182 . . . . . . . 8  |-  ( ( G  Fn  B  /\  <.
x ,  z >.  e.  G )  ->  x  e.  B )
31, 2anim12i 334 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  <. x ,  y
>.  e.  F )  /\  ( G  Fn  B  /\  <. x ,  z
>.  e.  G ) )  ->  ( x  e.  A  /\  x  e.  B ) )
43an4s 560 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  ->  ( x  e.  A  /\  x  e.  B ) )
5 elin 3223 . . . . . 6  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
64, 5sylibr 133 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  ->  x  e.  ( A  i^i  B ) )
7 vex 2658 . . . . . . . 8  |-  y  e. 
_V
87opres 4784 . . . . . . 7  |-  ( x  e.  ( A  i^i  B )  ->  ( <. x ,  y >.  e.  ( F  |`  ( A  i^i  B ) )  <->  <. x ,  y >.  e.  F
) )
9 vex 2658 . . . . . . . 8  |-  z  e. 
_V
109opres 4784 . . . . . . 7  |-  ( x  e.  ( A  i^i  B )  ->  ( <. x ,  z >.  e.  ( G  |`  ( A  i^i  B ) )  <->  <. x ,  z >.  e.  G
) )
118, 10anbi12d 462 . . . . . 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 157 . . . . 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 114 . . 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 4799 . . . 4  |-  ( F  |`  ( A  i^i  B
) )  C_  F
1716sseli 3057 . . 3  |-  ( <.
x ,  y >.  e.  ( F  |`  ( A  i^i  B ) )  ->  <. x ,  y
>.  e.  F )
18 resss 4799 . . . 4  |-  ( G  |`  ( A  i^i  B
) )  C_  G
1918sseli 3057 . . 3  |-  ( <.
x ,  z >.  e.  ( G  |`  ( A  i^i  B ) )  ->  <. x ,  z
>.  e.  G )
2017, 19anim12i 334 . 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 141 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 103    <-> wb 104    e. wcel 1461    i^i cin 3034   <.cop 3494    |` cres 4499    Fn wfn 5074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-xp 4503  df-rel 4504  df-dm 4507  df-res 4509  df-fun 5081  df-fn 5082
This theorem is referenced by: (None)
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