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Theorem dibdmN 31273
Description: Domain of the partial isomorphism A. (Contributed by NM, 8-Mar-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
dibdmN  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  {
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 dibdmN
StepHypRef Expression
1 dibfn.b . . 3  |-  B  =  ( Base `  K
)
2 dibfn.l . . 3  |-  .<_  =  ( le `  K )
3 dibfn.h . . 3  |-  H  =  ( LHyp `  K
)
4 dibfn.i . . 3  |-  I  =  ( ( DIsoB `  K
) `  W )
51, 2, 3, 4dibfnN 31272 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { x  e.  B  |  x  .<_  W } )
6 fndm 5485 . 2  |-  ( I  Fn  { x  e.  B  |  x  .<_  W }  ->  dom  I  =  { x  e.  B  |  x  .<_  W }
)
75, 6syl 16 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  {
x  e.  B  |  x  .<_  W } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2654   class class class wbr 4154   dom cdm 4819    Fn wfn 5390   ` cfv 5395   Basecbs 13397   lecple 13464   LHypclh 30099   DIsoBcdib 31254
This theorem is referenced by:  dibglbN  31282  dibintclN  31283  dihglblem3N  31411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-disoa 31145  df-dib 31255
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