Theorem mclsrcl 31458

Description: Reverse closure for the closure function. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mclsval.d	|- D = (mDV ` T )
mclsval.e	|- E = (mEx ` T )
mclsval.c	|- C = (mCls ` T )

Assertion

Ref	Expression
mclsrcl	|- ( A e. ( K C B ) -> ( T e. _V /\ K C_ D /\ B C_ E ) )

Proof of Theorem mclsrcl

Dummy variables h d t c m o p s x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0i 3920	. . 3 |- ( A e. ( K C B ) -> -. ( K C B ) = (/) )
2		mclsval.c	. . . . . 6 |- C = (mCls ` T )
3		fvprc 6185	. . . . . 6 |- ( -. T e. _V -> (mCls ` T ) = (/) )
4	2, 3	syl5eq 2668	. . . . 5 |- ( -. T e. _V -> C = (/) )
5	4	oveqd 6667	. . . 4 |- ( -. T e. _V -> ( K C B ) = ( K (/) B ) )
6		0ov 6682	. . . 4 |- ( K (/) B ) = (/)
7	5, 6	syl6eq 2672	. . 3 |- ( -. T e. _V -> ( K C B ) = (/) )
8	1, 7	nsyl2 142	. 2 |- ( A e. ( K C B ) -> T e. _V )
9		fveq2 6191	. . . . . . . . 9 |- ( t = T -> (mCls ` t ) = (mCls ` T ) )
10	9, 2	syl6eqr 2674	. . . . . . . 8 |- ( t = T -> (mCls ` t ) = C )
11	10	oveqd 6667	. . . . . . 7 |- ( t = T -> ( K (mCls ` t ) B ) = ( K C B ) )
12	11	eleq2d 2687	. . . . . 6 |- ( t = T -> ( A e. ( K (mCls ` t ) B ) <-> A e. ( K C B ) ) )
13		fvex 6201	. . . . . . . . 9 |- (mDV ` t ) e. _V
14	13	elpw2 4828	. . . . . . . 8 |- ( K e. ~P (mDV ` t ) <-> K C_ (mDV ` t ) )
15		fveq2 6191	. . . . . . . . . 10 |- ( t = T -> (mDV ` t ) = (mDV ` T ) )
16		mclsval.d	. . . . . . . . . 10 |- D = (mDV ` T )
17	15, 16	syl6eqr 2674	. . . . . . . . 9 |- ( t = T -> (mDV ` t ) = D )
18	17	sseq2d 3633	. . . . . . . 8 |- ( t = T -> ( K C_ (mDV ` t ) <-> K C_ D ) )
19	14, 18	syl5bb 272	. . . . . . 7 |- ( t = T -> ( K e. ~P (mDV ` t ) <-> K C_ D ) )
20		fvex 6201	. . . . . . . . 9 |- (mEx ` t ) e. _V
21	20	elpw2 4828	. . . . . . . 8 |- ( B e. ~P (mEx ` t ) <-> B C_ (mEx ` t ) )
22		fveq2 6191	. . . . . . . . . 10 |- ( t = T -> (mEx ` t ) = (mEx ` T ) )
23		mclsval.e	. . . . . . . . . 10 |- E = (mEx ` T )
24	22, 23	syl6eqr 2674	. . . . . . . . 9 |- ( t = T -> (mEx ` t ) = E )
25	24	sseq2d 3633	. . . . . . . 8 |- ( t = T -> ( B C_ (mEx ` t ) <-> B C_ E ) )
26	21, 25	syl5bb 272	. . . . . . 7 |- ( t = T -> ( B e. ~P (mEx ` t ) <-> B C_ E ) )
27	19, 26	anbi12d 747	. . . . . 6 |- ( t = T -> ( ( K e. ~P (mDV ` t ) /\ B e. ~P (mEx ` t ) ) <-> ( K C_ D /\ B C_ E ) ) )
28	12, 27	imbi12d 334	. . . . 5 |- ( t = T -> ( ( A e. ( K (mCls ` t ) B ) -> ( K e. ~P (mDV ` t ) /\ B e. ~P (mEx ` t ) ) ) <-> ( A e. ( K C B ) -> ( K C_ D /\ B C_ E ) ) ) )
29		vex 3203	. . . . . . 7 |- t e. _V
30	13	pwex 4848	. . . . . . . 8 |- ~P (mDV ` t ) e. _V
31	20	pwex 4848	. . . . . . . 8 |- ~P (mEx ` t ) e. _V
32	30, 31	mpt2ex 7247	. . . . . . 7 |- ( d e. ~P (mDV ` t ) , h e. ~P (mEx ` t ) |-> |^| { c | ( ( h u. ran (mVH ` t ) ) C_ c /\ A. m A. o A. p ( <. m , o , p >. e. (mAx ` t ) -> A. s e. ran (mSubst ` t ) ( ( ( s " ( o u. ran (mVH ` t ) ) ) C_ c /\ A. x A. y ( x m y -> ( ( (mVars ` t ) ` ( s ` ( (mVH ` t ) ` x ) ) ) X. ( (mVars ` t ) ` ( s ` ( (mVH ` t ) ` y ) ) ) ) C_ d ) ) -> ( s ` p ) e. c ) ) ) } ) e. _V
33		df-mcls 31394	. . . . . . . 8 |- mCls = ( t e. _V |-> ( d e. ~P (mDV ` t ) , h e. ~P (mEx ` t ) |-> |^| { c | ( ( h u. ran (mVH ` t ) ) C_ c /\ A. m A. o A. p ( <. m , o , p >. e. (mAx ` t ) -> A. s e. ran (mSubst ` t ) ( ( ( s " ( o u. ran (mVH ` t ) ) ) C_ c /\ A. x A. y ( x m y -> ( ( (mVars ` t ) ` ( s ` ( (mVH ` t ) ` x ) ) ) X. ( (mVars ` t ) ` ( s ` ( (mVH ` t ) ` y ) ) ) ) C_ d ) ) -> ( s ` p ) e. c ) ) ) } ) )
34	33	fvmpt2 6291	. . . . . . 7 |- ( ( t e. _V /\ ( d e. ~P (mDV ` t ) , h e. ~P (mEx ` t ) |-> |^| { c | ( ( h u. ran (mVH ` t ) ) C_ c /\ A. m A. o A. p ( <. m , o , p >. e. (mAx ` t ) -> A. s e. ran (mSubst ` t ) ( ( ( s " ( o u. ran (mVH ` t ) ) ) C_ c /\ A. x A. y ( x m y -> ( ( (mVars ` t ) ` ( s ` ( (mVH ` t ) ` x ) ) ) X. ( (mVars ` t ) ` ( s ` ( (mVH ` t ) ` y ) ) ) ) C_ d ) ) -> ( s ` p ) e. c ) ) ) } ) e. _V ) -> (mCls ` t ) = ( d e. ~P (mDV ` t ) , h e. ~P (mEx ` t ) |-> |^| { c | ( ( h u. ran (mVH ` t ) ) C_ c /\ A. m A. o A. p ( <. m , o , p >. e. (mAx ` t ) -> A. s e. ran (mSubst ` t ) ( ( ( s " ( o u. ran (mVH ` t ) ) ) C_ c /\ A. x A. y ( x m y -> ( ( (mVars ` t ) ` ( s ` ( (mVH ` t ) ` x ) ) ) X. ( (mVars ` t ) ` ( s ` ( (mVH ` t ) ` y ) ) ) ) C_ d ) ) -> ( s ` p ) e. c ) ) ) } ) )
35	29, 32, 34	mp2an 708	. . . . . 6 |- (mCls ` t ) = ( d e. ~P (mDV ` t ) , h e. ~P (mEx ` t ) |-> |^| { c | ( ( h u. ran (mVH ` t ) ) C_ c /\ A. m A. o A. p ( <. m , o , p >. e. (mAx ` t ) -> A. s e. ran (mSubst ` t ) ( ( ( s " ( o u. ran (mVH ` t ) ) ) C_ c /\ A. x A. y ( x m y -> ( ( (mVars ` t ) ` ( s ` ( (mVH ` t ) ` x ) ) ) X. ( (mVars ` t ) ` ( s ` ( (mVH ` t ) ` y ) ) ) ) C_ d ) ) -> ( s ` p ) e. c ) ) ) } )
36	35	elmpt2cl 6876	. . . . 5 |- ( A e. ( K (mCls ` t ) B ) -> ( K e. ~P (mDV ` t ) /\ B e. ~P (mEx ` t ) ) )
37	28, 36	vtoclg 3266	. . . 4 |- ( T e. _V -> ( A e. ( K C B ) -> ( K C_ D /\ B C_ E ) ) )
38	8, 37	mpcom 38	. . 3 |- ( A e. ( K C B ) -> ( K C_ D /\ B C_ E ) )
39	38	simpld 475	. 2 |- ( A e. ( K C B ) -> K C_ D )
40	38	simprd 479	. 2 |- ( A e. ( K C B ) -> B C_ E )
41	8, 39, 40	3jca 1242	1 |- ( A e. ( K C B ) -> ( T e. _V /\ K C_ D /\ B C_ E ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   _Vcvv 3200   u. cun 3572   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   <.cotp 4185   |^|cint 4475   class class class wbr 4653   X. cxp 5112   ran crn 5115   "cima 5117   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652  mAxcmax 31362  mExcmex 31364  mDVcmdv 31365  mVarscmvrs 31366  mSubstcmsub 31368  mVHcmvh 31369  mClscmcls 31374

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-mcls 31394

This theorem is referenced by: (None)