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Theorem bnd2lem 26193
Description: Lemma for equivbnd2 26194 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 5112 . . . . . 6  |-  ( M  |`  ( Y  X.  Y
) )  C_  M
31, 2eqsstri 3323 . . . . 5  |-  D  C_  M
4 dmss 5011 . . . . 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 26183 . . . . . 6  |-  ( D  e.  ( Bnd `  Y
)  ->  D  e.  ( Met `  Y ) )
7 metf 18271 . . . . . 6  |-  ( D  e.  ( Met `  Y
)  ->  D :
( Y  X.  Y
) --> RR )
8 fdm 5537 . . . . . 6  |-  ( D : ( Y  X.  Y ) --> RR  ->  dom 
D  =  ( Y  X.  Y ) )
96, 7, 83syl 19 . . . . 5  |-  ( D  e.  ( Bnd `  Y
)  ->  dom  D  =  ( Y  X.  Y
) )
109adantl 453 . . . 4  |-  ( ( M  e.  ( Met `  X )  /\  D  e.  ( Bnd `  Y
) )  ->  dom  D  =  ( Y  X.  Y ) )
11 metf 18271 . . . . . 6  |-  ( M  e.  ( Met `  X
)  ->  M :
( X  X.  X
) --> RR )
12 fdm 5537 . . . . . 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 452 . . . 4  |-  ( ( M  e.  ( Met `  X )  /\  D  e.  ( Bnd `  Y
) )  ->  dom  M  =  ( X  X.  X ) )
155, 10, 143sstr3d 3335 . . 3  |-  ( ( M  e.  ( Met `  X )  /\  D  e.  ( Bnd `  Y
) )  ->  ( Y  X.  Y )  C_  ( X  X.  X
) )
16 dmss 5011 . . 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 5031 . 2  |-  dom  ( Y  X.  Y )  =  Y
19 dmxpid 5031 . 2  |-  dom  ( X  X.  X )  =  X
2017, 18, 193sstr3g 3333 1  |-  ( ( M  e.  ( Met `  X )  /\  D  e.  ( Bnd `  Y
) )  ->  Y  C_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3265    X. cxp 4818   dom cdm 4820    |` cres 4822   -->wf 5392   ` cfv 5396   RRcr 8924   Metcme 16615   Bndcbnd 26169
This theorem is referenced by:  equivbnd2  26194  prdsbnd2  26197  cntotbnd  26198  cnpwstotbnd  26199
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-map 6958  df-met 16622  df-bnd 26181
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