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Theorem bnd2lem 29918
Description: Lemma for equivbnd2 29919 and similar theorems. (Contributed by Jeff Madsen, 16-Sep-2015.)
Hypothesis
Ref Expression
bnd2lem.1  |-  D  =  ( M  |`  ( Y  X.  Y ) )
Assertion
Ref Expression
bnd2lem  |-  ( ( M  e.  ( Met `  X )  /\  D  e.  ( Bnd `  Y
) )  ->  Y  C_  X )

Proof of Theorem bnd2lem
StepHypRef Expression
1 bnd2lem.1 . . . . . 6  |-  D  =  ( M  |`  ( Y  X.  Y ) )
2 resss 5297 . . . . . 6  |-  ( M  |`  ( Y  X.  Y
) )  C_  M
31, 2eqsstri 3534 . . . . 5  |-  D  C_  M
4 dmss 5202 . . . . 5  |-  ( D 
C_  M  ->  dom  D 
C_  dom  M )
53, 4mp1i 12 . . . 4  |-  ( ( M  e.  ( Met `  X )  /\  D  e.  ( Bnd `  Y
) )  ->  dom  D 
C_  dom  M )
6 bndmet 29908 . . . . . 6  |-  ( D  e.  ( Bnd `  Y
)  ->  D  e.  ( Met `  Y ) )
7 metf 20596 . . . . . 6  |-  ( D  e.  ( Met `  Y
)  ->  D :
( Y  X.  Y
) --> RR )
8 fdm 5735 . . . . . 6  |-  ( D : ( Y  X.  Y ) --> RR  ->  dom 
D  =  ( Y  X.  Y ) )
96, 7, 83syl 20 . . . . 5  |-  ( D  e.  ( Bnd `  Y
)  ->  dom  D  =  ( Y  X.  Y
) )
109adantl 466 . . . 4  |-  ( ( M  e.  ( Met `  X )  /\  D  e.  ( Bnd `  Y
) )  ->  dom  D  =  ( Y  X.  Y ) )
11 metf 20596 . . . . . 6  |-  ( M  e.  ( Met `  X
)  ->  M :
( X  X.  X
) --> RR )
12 fdm 5735 . . . . . 6  |-  ( M : ( X  X.  X ) --> RR  ->  dom 
M  =  ( X  X.  X ) )
1311, 12syl 16 . . . . 5  |-  ( M  e.  ( Met `  X
)  ->  dom  M  =  ( X  X.  X
) )
1413adantr 465 . . . 4  |-  ( ( M  e.  ( Met `  X )  /\  D  e.  ( Bnd `  Y
) )  ->  dom  M  =  ( X  X.  X ) )
155, 10, 143sstr3d 3546 . . 3  |-  ( ( M  e.  ( Met `  X )  /\  D  e.  ( Bnd `  Y
) )  ->  ( Y  X.  Y )  C_  ( X  X.  X
) )
16 dmss 5202 . . 3  |-  ( ( Y  X.  Y ) 
C_  ( X  X.  X )  ->  dom  ( Y  X.  Y
)  C_  dom  ( X  X.  X ) )
1715, 16syl 16 . 2  |-  ( ( M  e.  ( Met `  X )  /\  D  e.  ( Bnd `  Y
) )  ->  dom  ( Y  X.  Y
)  C_  dom  ( X  X.  X ) )
18 dmxpid 5222 . 2  |-  dom  ( Y  X.  Y )  =  Y
19 dmxpid 5222 . 2  |-  dom  ( X  X.  X )  =  X
2017, 18, 193sstr3g 3544 1  |-  ( ( M  e.  ( Met `  X )  /\  D  e.  ( Bnd `  Y
) )  ->  Y  C_  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3476    X. cxp 4997   dom cdm 4999    |` cres 5001   -->wf 5584   ` cfv 5588   RRcr 9491   Metcme 18203   Bndcbnd 29894
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549
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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-map 7422  df-met 18212  df-bnd 29906
This theorem is referenced by:  equivbnd2  29919  prdsbnd2  29922  cntotbnd  29923  cnpwstotbnd  29924
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