Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dibfnN Unicode version

Theorem dibfnN 31346
Description: Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dibfn.b  |-  B  =  ( Base `  K
)
dibfn.l  |-  .<_  =  ( le `  K )
dibfn.h  |-  H  =  ( LHyp `  K
)
dibfn.i  |-  I  =  ( ( DIsoB `  K
) `  W )
Assertion
Ref Expression
dibfnN  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { x  e.  B  |  x  .<_  W } )
Distinct variable groups:    x,  .<_    x, B    x, K    x, W
Allowed substitution hints:    H( x)    I( x)    V( x)

Proof of Theorem dibfnN
StepHypRef Expression
1 dibfn.h . . 3  |-  H  =  ( LHyp `  K
)
2 eqid 2283 . . 3  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
3 dibfn.i . . 3  |-  I  =  ( ( DIsoB `  K
) `  W )
41, 2, 3dibfna 31344 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  dom  (
( DIsoA `  K ) `  W ) )
5 dibfn.b . . . 4  |-  B  =  ( Base `  K
)
6 dibfn.l . . . 4  |-  .<_  =  ( le `  K )
75, 6, 1, 2diadm 31225 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  ( ( DIsoA `  K ) `  W
)  =  { x  e.  B  |  x  .<_  W } )
87fneq2d 5336 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( I  Fn  dom  ( ( DIsoA `  K
) `  W )  <->  I  Fn  { x  e.  B  |  x  .<_  W } ) )
94, 8mpbid 201 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { x  e.  B  |  x  .<_  W } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   class class class wbr 4023   dom cdm 4689    Fn wfn 5250   ` cfv 5255   Basecbs 13148   lecple 13215   LHypclh 30173   DIsoAcdia 31218   DIsoBcdib 31328
This theorem is referenced by:  dibdmN  31347  dibf11N  31351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-disoa 31219  df-dib 31329
  Copyright terms: Public domain W3C validator