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Theorem funpsstri 23289
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 5151 . . . . . 6  |-  ( ( Fun  H  /\  F  C_  H )  ->  ( H  |`  dom  F )  =  F )
21ex 425 . . . . 5  |-  ( Fun 
H  ->  ( F  C_  H  ->  ( H  |` 
dom  F )  =  F ) )
3 funssres 5151 . . . . . 6  |-  ( ( Fun  H  /\  G  C_  H )  ->  ( H  |`  dom  G )  =  G )
43ex 425 . . . . 5  |-  ( Fun 
H  ->  ( G  C_  H  ->  ( H  |` 
dom  G )  =  G ) )
52, 4anim12d 548 . . . 4  |-  ( Fun 
H  ->  ( ( F  C_  H  /\  G  C_  H )  ->  (
( H  |`  dom  F
)  =  F  /\  ( H  |`  dom  G
)  =  G ) ) )
6 ssres2 4889 . . . . . 6  |-  ( dom 
F  C_  dom  G  -> 
( H  |`  dom  F
)  C_  ( H  |` 
dom  G ) )
7 ssres2 4889 . . . . . 6  |-  ( dom 
G  C_  dom  F  -> 
( H  |`  dom  G
)  C_  ( H  |` 
dom  F ) )
86, 7orim12i 504 . . . . 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 3122 . . . . . 6  |-  ( ( ( H  |`  dom  F
)  =  F  /\  ( H  |`  dom  G
)  =  G )  ->  ( ( H  |`  dom  F )  C_  ( H  |`  dom  G
)  <->  F  C_  G ) )
10 sseq12 3122 . . . . . . 7  |-  ( ( ( H  |`  dom  G
)  =  G  /\  ( H  |`  dom  F
)  =  F )  ->  ( ( H  |`  dom  G )  C_  ( H  |`  dom  F
)  <->  G  C_  F ) )
1110ancoms 441 . . . . . 6  |-  ( ( ( H  |`  dom  F
)  =  F  /\  ( H  |`  dom  G
)  =  G )  ->  ( ( H  |`  dom  G )  C_  ( H  |`  dom  F
)  <->  G  C_  F ) )
129, 11orbi12d 693 . . . . 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 212 . . . 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 31 . . 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 1150 . 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 3198 . 2  |-  ( ( F  C_  G  \/  G  C_  F )  <->  ( F  C.  G  \/  F  =  G  \/  G  C.  F ) )
1715, 16sylib 190 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 set class
Syntax hints:    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    \/ w3o 938    /\ w3a 939    = wceq 1619    C_ wss 3078    C. wpss 3079   dom cdm 4580    |` cres 4582   Fun wfun 4586
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-res 4600  df-fun 4602
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