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Theorem dibfnN 31968
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 2296 . . 3  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
3 dibfn.i . . 3  |-  I  =  ( ( DIsoB `  K
) `  W )
41, 2, 3dibfna 31966 . 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 31847 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  ( ( DIsoA `  K ) `  W
)  =  { x  e.  B  |  x  .<_  W } )
87fneq2d 5352 . 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 1632    e. wcel 1696   {crab 2560   class class class wbr 4039   dom cdm 4705    Fn wfn 5266   ` cfv 5271   Basecbs 13164   lecple 13231   LHypclh 30795   DIsoAcdia 31840   DIsoBcdib 31950
This theorem is referenced by:  dibdmN  31969  dibf11N  31973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-disoa 31841  df-dib 31951
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