Theorem ntrclsfv1 38353

Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is a functional relation between them (Contributed by RP, 28-May-2021.)

Hypotheses

Ref	Expression
ntrcls.o	|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )
ntrcls.d	|- D = ( O ` B )
ntrcls.r	|- ( ph -> I D K )

Assertion

Ref	Expression
ntrclsfv1	|- ( ph -> ( D ` I ) = K )

Distinct variable groups:   B, i, j, k   ph, i, j, k

Allowed substitution hints:   D( i, j, k)   I( i, j, k)   K( i, j, k)   O( i, j, k)

Proof of Theorem ntrclsfv1

Step	Hyp	Ref	Expression
1		ntrcls.r	. 2 |- ( ph -> I D K )
2		ntrcls.o	. . . . . . 7 |- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )
3		ntrcls.d	. . . . . . 7 |- D = ( O ` B )
4	2, 3, 1	ntrclsf1o 38349	. . . . . 6 |- ( ph -> D : ( ~P B ^m ~P B ) -1-1-onto-> ( ~P B ^m ~P B ) )
5		f1ofn 6138	. . . . . 6 |- ( D : ( ~P B ^m ~P B ) -1-1-onto-> ( ~P B ^m ~P B ) -> D Fn ( ~P B ^m ~P B ) )
6	4, 5	syl 17	. . . . 5 |- ( ph -> D Fn ( ~P B ^m ~P B ) )
7	2, 3, 1	ntrclsiex 38351	. . . . 5 |- ( ph -> I e. ( ~P B ^m ~P B ) )
8	6, 7	jca 554	. . . 4 |- ( ph -> ( D Fn ( ~P B ^m ~P B ) /\ I e. ( ~P B ^m ~P B ) ) )
9		fnfun 5988	. . . . . 6 |- ( D Fn ( ~P B ^m ~P B ) -> Fun D )
10	9	adantr 481	. . . . 5 |- ( ( D Fn ( ~P B ^m ~P B ) /\ I e. ( ~P B ^m ~P B ) ) -> Fun D )
11		fndm 5990	. . . . . . 7 |- ( D Fn ( ~P B ^m ~P B ) -> dom D = ( ~P B ^m ~P B ) )
12	11	eleq2d 2687	. . . . . 6 |- ( D Fn ( ~P B ^m ~P B ) -> ( I e. dom D <-> I e. ( ~P B ^m ~P B ) ) )
13	12	biimpar 502	. . . . 5 |- ( ( D Fn ( ~P B ^m ~P B ) /\ I e. ( ~P B ^m ~P B ) ) -> I e. dom D )
14	10, 13	jca 554	. . . 4 |- ( ( D Fn ( ~P B ^m ~P B ) /\ I e. ( ~P B ^m ~P B ) ) -> ( Fun D /\ I e. dom D ) )
15	8, 14	syl 17	. . 3 |- ( ph -> ( Fun D /\ I e. dom D ) )
16		funbrfvb 6238	. . 3 |- ( ( Fun D /\ I e. dom D ) -> ( ( D ` I ) = K <-> I D K ) )
17	15, 16	syl 17	. 2 |- ( ph -> ( ( D ` I ) = K <-> I D K ) )
18	1, 17	mpbird 247	1 |- ( ph -> ( D ` I ) = K )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   Fun wfun 5882   Fn wfn 5883   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   ^m cmap 7857

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-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by:  ntrclsfv2  38354  ntrclscls00  38364  ntrclsiso  38365  ntrclsk2  38366  ntrclskb  38367  ntrclsk3  38368  ntrclsk13  38369