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Theorem ntrk0kbimka 38337
Description: If the interiors of disjoint sets are disjoint and the interior of the base set is the base set, then the interior of the empty set is the empty set. Obsolete version of ntrkbimka 38336. (Contributed by RP, 12-Jun-2021.)
Assertion
Ref Expression
ntrk0kbimka |- ( ( B e. V /\ I e. ( ~P B ^m ~P B ) ) -> ( ( ( I ` B ) = B /\ A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) ) -> ( I ` (/) ) = (/) ) )
Distinct variable groups:   B, s, t   I, s, t
Allowed substitution hints:   V( t, s)
Proof of Theorem ntrk0kbimka
StepHypRef Expression
1 pwidg 4173 . . . . 5 |- ( B e. V -> B e. ~P B )
21ad2antrr 762 . . . 4 |- ( ( ( B e. V /\ I e. ( ~P B ^m ~P B ) ) /\ ( ( I ` B ) = B /\ A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) ) ) -> B e. ~P B )
3 0elpw 4834 . . . . 5 |- (/) e. ~P B
43a1i 11 . . . 4 |- ( ( ( B e. V /\ I e. ( ~P B ^m ~P B ) ) /\ ( ( I ` B ) = B /\ A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) ) ) -> (/) e. ~P B )
5 simprr 796 . . . 4 |- ( ( ( B e. V /\ I e. ( ~P B ^m ~P B ) ) /\ ( ( I ` B ) = B /\ A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) ) ) -> A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) )
6 ineq1 3807 . . . . . . 7 |- ( s = B -> ( s i^i t ) = ( B i^i t ) )
76eqeq1d 2624 . . . . . 6 |- ( s = B -> ( ( s i^i t ) = (/) <-> ( B i^i t ) = (/) ) )
8 fveq2 6191 . . . . . . . 8 |- ( s = B -> ( I ` s ) = ( I ` B ) )
98ineq1d 3813 . . . . . . 7 |- ( s = B -> ( ( I ` s ) i^i ( I ` t ) ) = ( ( I ` B ) i^i ( I ` t ) ) )
109eqeq1d 2624 . . . . . 6 |- ( s = B -> ( ( ( I ` s ) i^i ( I ` t ) ) = (/) <-> ( ( I ` B ) i^i ( I ` t ) ) = (/) ) )
117, 10imbi12d 334 . . . . 5 |- ( s = B -> ( ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) <-> ( ( B i^i t ) = (/) -> ( ( I ` B ) i^i ( I ` t ) ) = (/) ) ) )
12 ineq2 3808 . . . . . . . 8 |- ( t = (/) -> ( B i^i t ) = ( B i^i (/) ) )
1312eqeq1d 2624 . . . . . . 7 |- ( t = (/) -> ( ( B i^i t ) = (/) <-> ( B i^i (/) ) = (/) ) )
14 fveq2 6191 . . . . . . . . 9 |- ( t = (/) -> ( I ` t ) = ( I ` (/) ) )
1514ineq2d 3814 . . . . . . . 8 |- ( t = (/) -> ( ( I ` B ) i^i ( I ` t ) ) = ( ( I ` B ) i^i ( I ` (/) ) ) )
1615eqeq1d 2624 . . . . . . 7 |- ( t = (/) -> ( ( ( I ` B ) i^i ( I ` t ) ) = (/) <-> ( ( I ` B ) i^i ( I ` (/) ) ) = (/) ) )
1713, 16imbi12d 334 . . . . . 6 |- ( t = (/) -> ( ( ( B i^i t ) = (/) -> ( ( I ` B ) i^i ( I ` t ) ) = (/) ) <-> ( ( B i^i (/) ) = (/) -> ( ( I ` B ) i^i ( I ` (/) ) ) = (/) ) ) )
18 in0 3968 . . . . . . 7 |- ( B i^i (/) ) = (/)
19 pm5.5 351 . . . . . . 7 |- ( ( B i^i (/) ) = (/) -> ( ( ( B i^i (/) ) = (/) -> ( ( I ` B ) i^i ( I ` (/) ) ) = (/) ) <-> ( ( I ` B ) i^i ( I ` (/) ) ) = (/) ) )
2018, 19mp1i 13 . . . . . 6 |- ( t = (/) -> ( ( ( B i^i (/) ) = (/) -> ( ( I ` B ) i^i ( I ` (/) ) ) = (/) ) <-> ( ( I ` B ) i^i ( I ` (/) ) ) = (/) ) )
2117, 20bitrd 268 . . . . 5 |- ( t = (/) -> ( ( ( B i^i t ) = (/) -> ( ( I ` B ) i^i ( I ` t ) ) = (/) ) <-> ( ( I ` B ) i^i ( I ` (/) ) ) = (/) ) )
2211, 21rspc2va 3323 . . . 4 |- ( ( ( B e. ~P B /\ (/) e. ~P B ) /\ A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) ) -> ( ( I ` B ) i^i ( I ` (/) ) ) = (/) )
232, 4, 5, 22syl21anc 1325 . . 3 |- ( ( ( B e. V /\ I e. ( ~P B ^m ~P B ) ) /\ ( ( I ` B ) = B /\ A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) ) ) -> ( ( I ` B ) i^i ( I ` (/) ) ) = (/) )
2423ex 450 . 2 |- ( ( B e. V /\ I e. ( ~P B ^m ~P B ) ) -> ( ( ( I ` B ) = B /\ A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) ) -> ( ( I ` B ) i^i ( I ` (/) ) ) = (/) ) )
25 elmapi 7879 . . . . . 6 |- ( I e. ( ~P B ^m ~P B ) -> I : ~P B --> ~P B )
2625adantl 482 . . . . 5 |- ( ( B e. V /\ I e. ( ~P B ^m ~P B ) ) -> I : ~P B --> ~P B )
273a1i 11 . . . . 5 |- ( ( B e. V /\ I e. ( ~P B ^m ~P B ) ) -> (/) e. ~P B )
2826, 27ffvelrnd 6360 . . . 4 |- ( ( B e. V /\ I e. ( ~P B ^m ~P B ) ) -> ( I ` (/) ) e. ~P B )
2928elpwid 4170 . . 3 |- ( ( B e. V /\ I e. ( ~P B ^m ~P B ) ) -> ( I ` (/) ) C_ B )
30 simpl 473 . . 3 |- ( ( ( I ` B ) = B /\ A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) ) -> ( I ` B ) = B )
31 ineq1 3807 . . . . . . . 8 |- ( ( I ` B ) = B -> ( ( I ` B ) i^i ( I ` (/) ) ) = ( B i^i ( I ` (/) ) ) )
32 incom 3805 . . . . . . . 8 |- ( B i^i ( I ` (/) ) ) = ( ( I ` (/) ) i^i B )
3331, 32syl6eq 2672 . . . . . . 7 |- ( ( I ` B ) = B -> ( ( I ` B ) i^i ( I ` (/) ) ) = ( ( I ` (/) ) i^i B ) )
3433eqeq1d 2624 . . . . . 6 |- ( ( I ` B ) = B -> ( ( ( I ` B ) i^i ( I ` (/) ) ) = (/) <-> ( ( I ` (/) ) i^i B ) = (/) ) )
3534biimpd 219 . . . . 5 |- ( ( I ` B ) = B -> ( ( ( I ` B ) i^i ( I ` (/) ) ) = (/) -> ( ( I ` (/) ) i^i B ) = (/) ) )
36 reldisj 4020 . . . . . . 7 |- ( ( I ` (/) ) C_ B -> ( ( ( I ` (/) ) i^i B ) = (/) <-> ( I ` (/) ) C_ ( B \ B ) ) )
3736biimpd 219 . . . . . 6 |- ( ( I ` (/) ) C_ B -> ( ( ( I ` (/) ) i^i B ) = (/) -> ( I ` (/) ) C_ ( B \ B ) ) )
38 difid 3948 . . . . . . . 8 |- ( B \ B ) = (/)
3938sseq2i 3630 . . . . . . 7 |- ( ( I ` (/) ) C_ ( B \ B ) <-> ( I ` (/) ) C_ (/) )
40 ss0 3974 . . . . . . 7 |- ( ( I ` (/) ) C_ (/) -> ( I ` (/) ) = (/) )
4139, 40sylbi 207 . . . . . 6 |- ( ( I ` (/) ) C_ ( B \ B ) -> ( I ` (/) ) = (/) )
4237, 41syl6com 37 . . . . 5 |- ( ( ( I ` (/) ) i^i B ) = (/) -> ( ( I ` (/) ) C_ B -> ( I ` (/) ) = (/) ) )
4335, 42syl6com 37 . . . 4 |- ( ( ( I ` B ) i^i ( I ` (/) ) ) = (/) -> ( ( I ` B ) = B -> ( ( I ` (/) ) C_ B -> ( I ` (/) ) = (/) ) ) )
4443com13 88 . . 3 |- ( ( I ` (/) ) C_ B -> ( ( I ` B ) = B -> ( ( ( I ` B ) i^i ( I ` (/) ) ) = (/) -> ( I ` (/) ) = (/) ) ) )
4529, 30, 44syl2im 40 . 2 |- ( ( B e. V /\ I e. ( ~P B ^m ~P B ) ) -> ( ( ( I ` B ) = B /\ A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) ) -> ( ( ( I ` B ) i^i ( I ` (/) ) ) = (/) -> ( I ` (/) ) = (/) ) ) )
4624, 45mpdd 43 1 |- ( ( B e. V /\ I e. ( ~P B ^m ~P B ) ) -> ( ( ( I ` B ) = B /\ A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) ) -> ( I ` (/) ) = (/) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   -->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-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-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-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)
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