Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  funpsstri Structured version   Unicode version

Theorem funpsstri 29122
Description: A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.)
Assertion
Ref Expression
funpsstri  |-  ( ( Fun  H  /\  ( F  C_  H  /\  G  C_  H )  /\  ( dom  F  C_  dom  G  \/  dom  G  C_  dom  F ) )  ->  ( F  C.  G  \/  F  =  G  \/  G  C.  F ) )

Proof of Theorem funpsstri
StepHypRef Expression
1 funssres 5634 . . . . . 6  |-  ( ( Fun  H  /\  F  C_  H )  ->  ( H  |`  dom  F )  =  F )
21ex 434 . . . . 5  |-  ( Fun 
H  ->  ( F  C_  H  ->  ( H  |` 
dom  F )  =  F ) )
3 funssres 5634 . . . . . 6  |-  ( ( Fun  H  /\  G  C_  H )  ->  ( H  |`  dom  G )  =  G )
43ex 434 . . . . 5  |-  ( Fun 
H  ->  ( G  C_  H  ->  ( H  |` 
dom  G )  =  G ) )
52, 4anim12d 563 . . . 4  |-  ( Fun 
H  ->  ( ( F  C_  H  /\  G  C_  H )  ->  (
( H  |`  dom  F
)  =  F  /\  ( H  |`  dom  G
)  =  G ) ) )
6 ssres2 5306 . . . . . 6  |-  ( dom 
F  C_  dom  G  -> 
( H  |`  dom  F
)  C_  ( H  |` 
dom  G ) )
7 ssres2 5306 . . . . . 6  |-  ( dom 
G  C_  dom  F  -> 
( H  |`  dom  G
)  C_  ( H  |` 
dom  F ) )
86, 7orim12i 516 . . . . 5  |-  ( ( dom  F  C_  dom  G  \/  dom  G  C_  dom  F )  ->  (
( H  |`  dom  F
)  C_  ( H  |` 
dom  G )  \/  ( H  |`  dom  G
)  C_  ( H  |` 
dom  F ) ) )
9 sseq12 3532 . . . . . 6  |-  ( ( ( H  |`  dom  F
)  =  F  /\  ( H  |`  dom  G
)  =  G )  ->  ( ( H  |`  dom  F )  C_  ( H  |`  dom  G
)  <->  F  C_  G ) )
10 sseq12 3532 . . . . . . 7  |-  ( ( ( H  |`  dom  G
)  =  G  /\  ( H  |`  dom  F
)  =  F )  ->  ( ( H  |`  dom  G )  C_  ( H  |`  dom  F
)  <->  G  C_  F ) )
1110ancoms 453 . . . . . 6  |-  ( ( ( H  |`  dom  F
)  =  F  /\  ( H  |`  dom  G
)  =  G )  ->  ( ( H  |`  dom  G )  C_  ( H  |`  dom  F
)  <->  G  C_  F ) )
129, 11orbi12d 709 . . . . 5  |-  ( ( ( H  |`  dom  F
)  =  F  /\  ( H  |`  dom  G
)  =  G )  ->  ( ( ( H  |`  dom  F ) 
C_  ( H  |`  dom  G )  \/  ( H  |`  dom  G ) 
C_  ( H  |`  dom  F ) )  <->  ( F  C_  G  \/  G  C_  F ) ) )
138, 12syl5ib 219 . . . 4  |-  ( ( ( H  |`  dom  F
)  =  F  /\  ( H  |`  dom  G
)  =  G )  ->  ( ( dom 
F  C_  dom  G  \/  dom  G  C_  dom  F )  ->  ( F  C_  G  \/  G  C_  F
) ) )
145, 13syl6 33 . . 3  |-  ( Fun 
H  ->  ( ( F  C_  H  /\  G  C_  H )  ->  (
( dom  F  C_  dom  G  \/  dom  G  C_  dom  F )  ->  ( F  C_  G  \/  G  C_  F ) ) ) )
15143imp 1190 . 2  |-  ( ( Fun  H  /\  ( F  C_  H  /\  G  C_  H )  /\  ( dom  F  C_  dom  G  \/  dom  G  C_  dom  F ) )  ->  ( F  C_  G  \/  G  C_  F ) )
16 sspsstri 3611 . 2  |-  ( ( F  C_  G  \/  G  C_  F )  <->  ( F  C.  G  \/  F  =  G  \/  G  C.  F ) )
1715, 16sylib 196 1  |-  ( ( Fun  H  /\  ( F  C_  H  /\  G  C_  H )  /\  ( dom  F  C_  dom  G  \/  dom  G  C_  dom  F ) )  ->  ( F  C.  G  \/  F  =  G  \/  G  C.  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 972    /\ w3a 973    = wceq 1379    C_ wss 3481    C. wpss 3482   dom cdm 5005    |` cres 5007   Fun wfun 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-res 5017  df-fun 5596
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator