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Theorem funsseq 29442
Description: Given two functions with equal domains, equality only requires one direction of the subset relationship. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by Mario Carneiro, 3-May-2015.)
Assertion
Ref Expression
funsseq  |-  ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  -> 
( F  =  G  <-> 
F  C_  G )
)

Proof of Theorem funsseq
StepHypRef Expression
1 eqimss 3541 . 2  |-  ( F  =  G  ->  F  C_  G )
2 simpl3 999 . . . . 5  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  ->  dom  F  =  dom  G
)
32reseq2d 5262 . . . 4  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  -> 
( G  |`  dom  F
)  =  ( G  |`  dom  G ) )
4 funssres 5610 . . . . 5  |-  ( ( Fun  G  /\  F  C_  G )  ->  ( G  |`  dom  F )  =  F )
543ad2antl2 1157 . . . 4  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  -> 
( G  |`  dom  F
)  =  F )
6 simpl2 998 . . . . 5  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  ->  Fun  G )
7 funrel 5587 . . . . 5  |-  ( Fun 
G  ->  Rel  G )
8 resdm 5303 . . . . 5  |-  ( Rel 
G  ->  ( G  |` 
dom  G )  =  G )
96, 7, 83syl 20 . . . 4  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  -> 
( G  |`  dom  G
)  =  G )
103, 5, 93eqtr3d 2503 . . 3  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  ->  F  =  G )
1110ex 432 . 2  |-  ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  -> 
( F  C_  G  ->  F  =  G ) )
121, 11impbid2 204 1  |-  ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  -> 
( F  =  G  <-> 
F  C_  G )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    C_ wss 3461   dom cdm 4988    |` cres 4990   Rel wrel 4993   Fun wfun 5564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-res 5000  df-fun 5572
This theorem is referenced by: (None)
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