Theorem ntrrn 38420

Description: The range of the interior function of a topology a subset of the open sets of the topology. (Contributed by RP, 22-Apr-2021.)

Hypotheses

Ref	Expression
ntrrn.x	|- X = U. J
ntrrn.i	|- I = ( int ` J )

Assertion

Ref	Expression
ntrrn	|- ( J e. Top -> ran I C_ J )

Proof of Theorem ntrrn

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ntrrn.i	. . 3 |- I = ( int ` J )
2	1	rneqi 5352	. 2 |- ran I = ran ( int ` J )
3		vpwex 4849	. . . . . . . 8 |- ~P s e. _V
4	3	inex2 4800	. . . . . . 7 |- ( J i^i ~P s ) e. _V
5	4	uniex 6953	. . . . . 6 |- U. ( J i^i ~P s ) e. _V
6	5	rgenw 2924	. . . . 5 |- A. s e. ~P X U. ( J i^i ~P s ) e. _V
7		nfcv 2764	. . . . . 6 |- F/_ s ~P X
8	7	fnmptf 6016	. . . . 5 |- ( A. s e. ~P X U. ( J i^i ~P s ) e. _V -> ( s e. ~P X |-> U. ( J i^i ~P s ) ) Fn ~P X )
9	6, 8	mp1i 13	. . . 4 |- ( J e. Top -> ( s e. ~P X |-> U. ( J i^i ~P s ) ) Fn ~P X )
10		ntrrn.x	. . . . . 6 |- X = U. J
11	10	ntrfval 20828	. . . . 5 |- ( J e. Top -> ( int ` J ) = ( s e. ~P X |-> U. ( J i^i ~P s ) ) )
12	11	fneq1d 5981	. . . 4 |- ( J e. Top -> ( ( int ` J ) Fn ~P X <-> ( s e. ~P X |-> U. ( J i^i ~P s ) ) Fn ~P X ) )
13	9, 12	mpbird 247	. . 3 |- ( J e. Top -> ( int ` J ) Fn ~P X )
14		elpwi 4168	. . . . 5 |- ( s e. ~P X -> s C_ X )
15	10	ntropn 20853	. . . . . 6 |- ( ( J e. Top /\ s C_ X ) -> ( ( int ` J ) ` s ) e. J )
16	15	ex 450	. . . . 5 |- ( J e. Top -> ( s C_ X -> ( ( int ` J ) ` s ) e. J ) )
17	14, 16	syl5 34	. . . 4 |- ( J e. Top -> ( s e. ~P X -> ( ( int ` J ) ` s ) e. J ) )
18	17	ralrimiv 2965	. . 3 |- ( J e. Top -> A. s e. ~P X ( ( int ` J ) ` s ) e. J )
19		fnfvrnss 6390	. . 3 |- ( ( ( int ` J ) Fn ~P X /\ A. s e. ~P X ( ( int ` J ) ` s ) e. J ) -> ran ( int ` J ) C_ J )
20	13, 18, 19	syl2anc 693	. 2 |- ( J e. Top -> ran ( int ` J ) C_ J )
21	2, 20	syl5eqss 3649	1 |- ( J e. Top -> ran I C_ J )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   |-> cmpt 4729   ran crn 5115   Fn wfn 5883   ` cfv 5888   Topctop 20698   intcnt 20821

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-top 20699  df-ntr 20824

This theorem is referenced by:  ntrf  38421