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Theorem bnd2lem 28688
Description: Lemma for equivbnd2 28689 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 5133 . . . . . 6  |-  ( M  |`  ( Y  X.  Y
) )  C_  M
31, 2eqsstri 3385 . . . . 5  |-  D  C_  M
4 dmss 5038 . . . . 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 28678 . . . . . 6  |-  ( D  e.  ( Bnd `  Y
)  ->  D  e.  ( Met `  Y ) )
7 metf 19904 . . . . . 6  |-  ( D  e.  ( Met `  Y
)  ->  D :
( Y  X.  Y
) --> RR )
8 fdm 5562 . . . . . 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 19904 . . . . . 6  |-  ( M  e.  ( Met `  X
)  ->  M :
( X  X.  X
) --> RR )
12 fdm 5562 . . . . . 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 3397 . . 3  |-  ( ( M  e.  ( Met `  X )  /\  D  e.  ( Bnd `  Y
) )  ->  ( Y  X.  Y )  C_  ( X  X.  X
) )
16 dmss 5038 . . 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 5058 . 2  |-  dom  ( Y  X.  Y )  =  Y
19 dmxpid 5058 . 2  |-  dom  ( X  X.  X )  =  X
2017, 18, 193sstr3g 3395 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 1369    e. wcel 1756    C_ wss 3327    X. cxp 4837   dom cdm 4839    |` cres 4841   -->wf 5413   ` cfv 5417   RRcr 9280   Metcme 17801   Bndcbnd 28664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-map 7215  df-met 17810  df-bnd 28676
This theorem is referenced by:  equivbnd2  28689  prdsbnd2  28692  cntotbnd  28693  cnpwstotbnd  28694
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