Theorem dicdmN 36473

Description: Domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dicfn.l	|- .<_ = ( le ` K )
dicfn.a	|- A = ( Atoms ` K )
dicfn.h	|- H = ( LHyp ` K )
dicfn.i	|- I = ( ( DIsoC ` K ) ` W )

Assertion

Ref	Expression
dicdmN	|- ( ( K e. V /\ W e. H ) -> dom I = { p e. A | -. p .<_ W } )

Distinct variable groups:   .<_ , p   A, p   K, p   W, p

Allowed substitution hints:   H( p)   I( p)   V( p)

Proof of Theorem dicdmN

Step	Hyp	Ref	Expression
1		dicfn.l	. . 3 |- .<_ = ( le ` K )
2		dicfn.a	. . 3 |- A = ( Atoms ` K )
3		dicfn.h	. . 3 |- H = ( LHyp ` K )
4		dicfn.i	. . 3 |- I = ( ( DIsoC ` K ) ` W )
5	1, 2, 3, 4	dicfnN 36472	. 2 |- ( ( K e. V /\ W e. H ) -> I Fn { p e. A | -. p .<_ W } )
6		fndm 5990	. 2 |- ( I Fn { p e. A | -. p .<_ W } -> dom I = { p e. A | -. p .<_ W } )
7	5, 6	syl 17	1 |- ( ( K e. V /\ W e. H ) -> dom I = { p e. A | -. p .<_ W } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   class class class wbr 4653   dom cdm 5114   Fn wfn 5883   ` cfv 5888   lecple 15948   Atomscatm 34550   LHypclh 35270   DIsoCcdic 36461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-dic 36462

This theorem is referenced by:  dicvalrelN  36474