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Theorem dibdmN 34434
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 34433 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { x  e.  B  |  x  .<_  W } )
6 fndm 5693 . 2  |-  ( I  Fn  { x  e.  B  |  x  .<_  W }  ->  dom  I  =  { x  e.  B  |  x  .<_  W }
)
75, 6syl 17 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  {
x  e.  B  |  x  .<_  W } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   {crab 2786   class class class wbr 4426   dom cdm 4854    Fn wfn 5596   ` cfv 5601   Basecbs 15084   lecple 15159   LHypclh 33258   DIsoBcdib 34415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-disoa 34306  df-dib 34416
This theorem is referenced by:  dibglbN  34443  dibintclN  34444  dihglblem3N  34572
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