Theorem ntrclsss 38361

Description: If interior and closure functions are related then a subset relation of a pair of function values is equivalent to subset relation of a pair of the other function's values. (Contributed by RP, 27-Jun-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 )
ntrclsfv.s	|- ( ph -> S e. ~P B )
ntrclsfv.t	|- ( ph -> T e. ~P B )

Assertion

Ref	Expression
ntrclsss	|- ( ph -> ( ( I ` S ) C_ ( I ` T ) <-> ( K ` ( B \ T ) ) C_ ( K ` ( B \ S ) ) ) )

Distinct variable groups:   B, i, j, k   j, K, k   S, j   T, j   ph, i, j, k

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

Proof of Theorem ntrclsss

Step	Hyp	Ref	Expression
1		ntrcls.o	. . . 4 |- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )
2		ntrcls.d	. . . 4 |- D = ( O ` B )
3		ntrcls.r	. . . 4 |- ( ph -> I D K )
4		ntrclsfv.s	. . . 4 |- ( ph -> S e. ~P B )
5	1, 2, 3, 4	ntrclsfv 38357	. . 3 |- ( ph -> ( I ` S ) = ( B \ ( K ` ( B \ S ) ) ) )
6		ntrclsfv.t	. . . 4 |- ( ph -> T e. ~P B )
7	1, 2, 3, 6	ntrclsfv 38357	. . 3 |- ( ph -> ( I ` T ) = ( B \ ( K ` ( B \ T ) ) ) )
8	5, 7	sseq12d 3634	. 2 |- ( ph -> ( ( I ` S ) C_ ( I ` T ) <-> ( B \ ( K ` ( B \ S ) ) ) C_ ( B \ ( K ` ( B \ T ) ) ) ) )
9	1, 2, 3	ntrclskex 38352	. . . 4 |- ( ph -> K e. ( ~P B ^m ~P B ) )
10	9	ancli 574	. . 3 |- ( ph -> ( ph /\ K e. ( ~P B ^m ~P B ) ) )
11		elmapi 7879	. . . . . . 7 |- ( K e. ( ~P B ^m ~P B ) -> K : ~P B --> ~P B )
12	11	adantl 482	. . . . . 6 |- ( ( ph /\ K e. ( ~P B ^m ~P B ) ) -> K : ~P B --> ~P B )
13	2, 3	ntrclsrcomplex 38333	. . . . . . 7 |- ( ph -> ( B \ T ) e. ~P B )
14	13	adantr 481	. . . . . 6 |- ( ( ph /\ K e. ( ~P B ^m ~P B ) ) -> ( B \ T ) e. ~P B )
15	12, 14	ffvelrnd 6360	. . . . 5 |- ( ( ph /\ K e. ( ~P B ^m ~P B ) ) -> ( K ` ( B \ T ) ) e. ~P B )
16	15	elpwid 4170	. . . 4 |- ( ( ph /\ K e. ( ~P B ^m ~P B ) ) -> ( K ` ( B \ T ) ) C_ B )
17	2, 3	ntrclsrcomplex 38333	. . . . . . 7 |- ( ph -> ( B \ S ) e. ~P B )
18	17	adantr 481	. . . . . 6 |- ( ( ph /\ K e. ( ~P B ^m ~P B ) ) -> ( B \ S ) e. ~P B )
19	12, 18	ffvelrnd 6360	. . . . 5 |- ( ( ph /\ K e. ( ~P B ^m ~P B ) ) -> ( K ` ( B \ S ) ) e. ~P B )
20	19	elpwid 4170	. . . 4 |- ( ( ph /\ K e. ( ~P B ^m ~P B ) ) -> ( K ` ( B \ S ) ) C_ B )
21	16, 20	jca 554	. . 3 |- ( ( ph /\ K e. ( ~P B ^m ~P B ) ) -> ( ( K ` ( B \ T ) ) C_ B /\ ( K ` ( B \ S ) ) C_ B ) )
22		sscon34b 38317	. . 3 |- ( ( ( K ` ( B \ T ) ) C_ B /\ ( K ` ( B \ S ) ) C_ B ) -> ( ( K ` ( B \ T ) ) C_ ( K ` ( B \ S ) ) <-> ( B \ ( K ` ( B \ S ) ) ) C_ ( B \ ( K ` ( B \ T ) ) ) ) )
23	10, 21, 22	3syl 18	. 2 |- ( ph -> ( ( K ` ( B \ T ) ) C_ ( K ` ( B \ S ) ) <-> ( B \ ( K ` ( B \ S ) ) ) C_ ( B \ ( K ` ( B \ T ) ) ) ) )
24	8, 23	bitr4d 271	1 |- ( ph -> ( ( I ` S ) C_ ( I ` T ) <-> ( K ` ( B \ T ) ) C_ ( K ` ( B \ S ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` 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: (None)