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Theorem dibdmN 35972
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 35971 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { x  e.  B  |  x  .<_  W } )
6 fndm 5680 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818   class class class wbr 4447   dom cdm 4999    Fn wfn 5583   ` cfv 5588   Basecbs 14490   lecple 14562   LHypclh 34798   DIsoBcdib 35953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-disoa 35844  df-dib 35954
This theorem is referenced by:  dibglbN  35981  dibintclN  35982  dihglblem3N  36110
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