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Theorem funsseq 29126
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 3561 . 2  |-  ( F  =  G  ->  F  C_  G )
2 simpl3 1001 . . . . 5  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  ->  dom  F  =  dom  G
)
32reseq2d 5279 . . . 4  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  -> 
( G  |`  dom  F
)  =  ( G  |`  dom  G ) )
4 funssres 5634 . . . . 5  |-  ( ( Fun  G  /\  F  C_  G )  ->  ( G  |`  dom  F )  =  F )
543ad2antl2 1159 . . . 4  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  -> 
( G  |`  dom  F
)  =  F )
6 simpl2 1000 . . . . 5  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  ->  Fun  G )
7 funrel 5611 . . . . 5  |-  ( Fun 
G  ->  Rel  G )
8 resdm 5321 . . . . 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 2516 . . 3  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  ->  F  =  G )
1110ex 434 . 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 369    /\ w3a 973    = wceq 1379    C_ wss 3481   dom cdm 5005    |` cres 5007   Rel wrel 5010   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-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-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)
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