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Theorem dibfnN 35159
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 2454 . . 3  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
3 dibfn.i . . 3  |-  I  =  ( ( DIsoB `  K
) `  W )
41, 2, 3dibfna 35157 . 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 35038 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  ( ( DIsoA `  K ) `  W
)  =  { x  e.  B  |  x  .<_  W } )
87fneq2d 5613 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( I  Fn  dom  ( ( DIsoA `  K
) `  W )  <->  I  Fn  { x  e.  B  |  x  .<_  W } ) )
94, 8mpbid 210 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { x  e.  B  |  x  .<_  W } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   {crab 2803   class class class wbr 4403   dom cdm 4951    Fn wfn 5524   ` cfv 5529   Basecbs 14295   lecple 14367   LHypclh 33986   DIsoAcdia 35031   DIsoBcdib 35141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-disoa 35032  df-dib 35142
This theorem is referenced by:  dibdmN  35160  dibf11N  35164
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