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Theorem 2elresin 5631
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 5623 . . . . . . . 8  |-  ( ( F  Fn  A  /\  <.
x ,  y >.  e.  F )  ->  x  e.  A )
2 fnop 5623 . . . . . . . 8  |-  ( ( G  Fn  B  /\  <.
x ,  z >.  e.  G )  ->  x  e.  B )
31, 2anim12i 566 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  <. x ,  y
>.  e.  F )  /\  ( G  Fn  B  /\  <. x ,  z
>.  e.  G ) )  ->  ( x  e.  A  /\  x  e.  B ) )
43an4s 822 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  ->  ( x  e.  A  /\  x  e.  B ) )
5 elin 3648 . . . . . 6  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
64, 5sylibr 212 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  ->  x  e.  ( A  i^i  B ) )
7 vex 3081 . . . . . . . 8  |-  y  e. 
_V
87opres 5229 . . . . . . 7  |-  ( x  e.  ( A  i^i  B )  ->  ( <. x ,  y >.  e.  ( F  |`  ( A  i^i  B ) )  <->  <. x ,  y >.  e.  F
) )
9 vex 3081 . . . . . . . 8  |-  z  e. 
_V
109opres 5229 . . . . . . 7  |-  ( x  e.  ( A  i^i  B )  ->  ( <. x ,  z >.  e.  ( G  |`  ( A  i^i  B ) )  <->  <. x ,  z >.  e.  G
) )
118, 10anbi12d 710 . . . . . 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 223 . . . . 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 16 . . . 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 434 . . 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 48 . 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 5243 . . . 4  |-  ( F  |`  ( A  i^i  B
) )  C_  F
1716sseli 3461 . . 3  |-  ( <.
x ,  y >.  e.  ( F  |`  ( A  i^i  B ) )  ->  <. x ,  y
>.  e.  F )
18 resss 5243 . . . 4  |-  ( G  |`  ( A  i^i  B
) )  C_  G
1918sseli 3461 . . 3  |-  ( <.
x ,  z >.  e.  ( G  |`  ( A  i^i  B ) )  ->  <. x ,  z
>.  e.  G )
2017, 19anim12i 566 . 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 203 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1758    i^i cin 3436   <.cop 3992    |` cres 4951    Fn wfn 5522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-xp 4955  df-rel 4956  df-dm 4959  df-res 4961  df-fun 5529  df-fn 5530
This theorem is referenced by: (None)
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