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Theorem funpsstri 23532
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 5294 . . . . . 6  |-  ( ( Fun  H  /\  F  C_  H )  ->  ( H  |`  dom  F )  =  F )
21ex 423 . . . . 5  |-  ( Fun 
H  ->  ( F  C_  H  ->  ( H  |` 
dom  F )  =  F ) )
3 funssres 5294 . . . . . 6  |-  ( ( Fun  H  /\  G  C_  H )  ->  ( H  |`  dom  G )  =  G )
43ex 423 . . . . 5  |-  ( Fun 
H  ->  ( G  C_  H  ->  ( H  |` 
dom  G )  =  G ) )
52, 4anim12d 546 . . . 4  |-  ( Fun 
H  ->  ( ( F  C_  H  /\  G  C_  H )  ->  (
( H  |`  dom  F
)  =  F  /\  ( H  |`  dom  G
)  =  G ) ) )
6 ssres2 4982 . . . . . 6  |-  ( dom 
F  C_  dom  G  -> 
( H  |`  dom  F
)  C_  ( H  |` 
dom  G ) )
7 ssres2 4982 . . . . . 6  |-  ( dom 
G  C_  dom  F  -> 
( H  |`  dom  G
)  C_  ( H  |` 
dom  F ) )
86, 7orim12i 502 . . . . 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 3201 . . . . . 6  |-  ( ( ( H  |`  dom  F
)  =  F  /\  ( H  |`  dom  G
)  =  G )  ->  ( ( H  |`  dom  F )  C_  ( H  |`  dom  G
)  <->  F  C_  G ) )
10 sseq12 3201 . . . . . . 7  |-  ( ( ( H  |`  dom  G
)  =  G  /\  ( H  |`  dom  F
)  =  F )  ->  ( ( H  |`  dom  G )  C_  ( H  |`  dom  F
)  <->  G  C_  F ) )
1110ancoms 439 . . . . . 6  |-  ( ( ( H  |`  dom  F
)  =  F  /\  ( H  |`  dom  G
)  =  G )  ->  ( ( H  |`  dom  G )  C_  ( H  |`  dom  F
)  <->  G  C_  F ) )
129, 11orbi12d 690 . . . . 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 210 . . . 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 29 . . 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 1145 . 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 3278 . 2  |-  ( ( F  C_  G  \/  G  C_  F )  <->  ( F  C.  G  \/  F  =  G  \/  G  C.  F ) )
1715, 16sylib 188 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 4    <-> wb 176    \/ wo 357    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1623    C_ wss 3152    C. wpss 3153   dom cdm 4689    |` cres 4691   Fun wfun 5249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-fun 5257
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