Theorem cnntri 15251

Description: Property of the preimage of an interior. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
cncls2i.1	|- Y = U. K

Assertion

Ref	Expression
cnntri	|- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( `' F " ( ( int ` K ) ` S ) ) C_ ( ( int ` J ) ` ( `' F " S ) ) )

Proof of Theorem cnntri

Step	Hyp	Ref	Expression
1		cntop1 15228	. . 3 |- ( F e. ( J Cn K ) -> J e. Top )
2	1	adantr 276	. 2 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> J e. Top )
3		cnvimass 5148	. . 3 |- ( `' F " S ) C_ dom F
4		eqid 2238	. . . . . 6 |- U. J = U. J
5		cncls2i.1	. . . . . 6 |- Y = U. K
6	4, 5	cnf 15231	. . . . 5 |- ( F e. ( J Cn K ) -> F : U. J --> Y )
7	6	fdmd 5538	. . . 4 |- ( F e. ( J Cn K ) -> dom F = U. J )
8	7	adantr 276	. . 3 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> dom F = U. J )
9	3, 8	sseqtrid 3298	. 2 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( `' F " S ) C_ U. J )
10		cntop2 15229	. . . 4 |- ( F e. ( J Cn K ) -> K e. Top )
11	5	ntropn 15144	. . . 4 |- ( ( K e. Top /\ S C_ Y ) -> ( ( int ` K ) ` S ) e. K )
12	10, 11	sylan 283	. . 3 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( ( int ` K ) ` S ) e. K )
13		cnima 15247	. . 3 |- ( ( F e. ( J Cn K ) /\ ( ( int ` K ) ` S ) e. K ) -> ( `' F " ( ( int ` K ) ` S ) ) e. J )
14	12, 13	syldan 282	. 2 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( `' F " ( ( int ` K ) ` S ) ) e. J )
15	5	ntrss2 15148	. . . 4 |- ( ( K e. Top /\ S C_ Y ) -> ( ( int ` K ) ` S ) C_ S )
16	10, 15	sylan 283	. . 3 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( ( int ` K ) ` S ) C_ S )
17		imass2 5161	. . 3 |- ( ( ( int ` K ) ` S ) C_ S -> ( `' F " ( ( int ` K ) ` S ) ) C_ ( `' F " S ) )
18	16, 17	syl 14	. 2 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( `' F " ( ( int ` K ) ` S ) ) C_ ( `' F " S ) )
19	4	ssntr 15149	. 2 |- ( ( ( J e. Top /\ ( `' F " S ) C_ U. J ) /\ ( ( `' F " ( ( int ` K ) ` S ) ) e. J /\ ( `' F " ( ( int ` K ) ` S ) ) C_ ( `' F " S ) ) ) -> ( `' F " ( ( int ` K ) ` S ) ) C_ ( ( int ` J ) ` ( `' F " S ) ) )
20	2, 9, 14, 18, 19	syl22anc 1279	1 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( `' F " ( ( int ` K ) ` S ) ) C_ ( ( int ` J ) ` ( `' F " S ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   C_ wss 3220   U.cuni 3933   `'ccnv 4771   dom cdm 4772   "cima 4775   ` cfv 5375  (class class class)co 6078   Topctop 15024   intcnt 15120   Cn ccn 15212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6081  df-oprab 6082  df-mpo 6083  df-1st 6367  df-2nd 6368  df-map 6917  df-top 15025  df-topon 15038  df-ntr 15123  df-cn 15215

This theorem is referenced by:  cnntr  15252  hmeontr  15340