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Theorem bnd2lem 26481
Description: Lemma for equivbnd2 26482 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 5162 . . . . . 6  |-  ( M  |`  ( Y  X.  Y
) )  C_  M
31, 2eqsstri 3370 . . . . 5  |-  D  C_  M
4 dmss 5061 . . . . 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 26471 . . . . . 6  |-  ( D  e.  ( Bnd `  Y
)  ->  D  e.  ( Met `  Y ) )
7 metf 18352 . . . . . 6  |-  ( D  e.  ( Met `  Y
)  ->  D :
( Y  X.  Y
) --> RR )
8 fdm 5587 . . . . . 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 18352 . . . . . 6  |-  ( M  e.  ( Met `  X
)  ->  M :
( X  X.  X
) --> RR )
12 fdm 5587 . . . . . 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 3382 . . 3  |-  ( ( M  e.  ( Met `  X )  /\  D  e.  ( Bnd `  Y
) )  ->  ( Y  X.  Y )  C_  ( X  X.  X
) )
16 dmss 5061 . . 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 5081 . 2  |-  dom  ( Y  X.  Y )  =  Y
19 dmxpid 5081 . 2  |-  dom  ( X  X.  X )  =  X
2017, 18, 193sstr3g 3380 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 1652    e. wcel 1725    C_ wss 3312    X. cxp 4868   dom cdm 4870    |` cres 4872   -->wf 5442   ` cfv 5446   RRcr 8981   Metcme 16679   Bndcbnd 26457
This theorem is referenced by:  equivbnd2  26482  prdsbnd2  26485  cntotbnd  26486  cnpwstotbnd  26487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-met 16688  df-bnd 26469
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