Theorem mrcss 16276

Description: Closure preserves subset ordering. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Hypothesis

Ref	Expression
mrcfval.f	|- F = (mrCls ` C )

Assertion

Ref	Expression
mrcss	|- ( ( C e. (Moore ` X ) /\ U C_ V /\ V C_ X ) -> ( F ` U ) C_ ( F ` V ) )

Proof of Theorem mrcss

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sstr2 3610	. . . . . 6 |- ( U C_ V -> ( V C_ s -> U C_ s ) )
2	1	adantr 481	. . . . 5 |- ( ( U C_ V /\ s e. C ) -> ( V C_ s -> U C_ s ) )
3	2	ss2rabdv 3683	. . . 4 |- ( U C_ V -> { s e. C | V C_ s } C_ { s e. C | U C_ s } )
4		intss 4498	. . . 4 |- ( { s e. C | V C_ s } C_ { s e. C | U C_ s } -> |^| { s e. C | U C_ s } C_ |^| { s e. C | V C_ s } )
5	3, 4	syl 17	. . 3 |- ( U C_ V -> |^| { s e. C | U C_ s } C_ |^| { s e. C | V C_ s } )
6	5	3ad2ant2 1083	. 2 |- ( ( C e. (Moore ` X ) /\ U C_ V /\ V C_ X ) -> |^| { s e. C | U C_ s } C_ |^| { s e. C | V C_ s } )
7		simp1 1061	. . 3 |- ( ( C e. (Moore ` X ) /\ U C_ V /\ V C_ X ) -> C e. (Moore ` X ) )
8		sstr 3611	. . . 4 |- ( ( U C_ V /\ V C_ X ) -> U C_ X )
9	8	3adant1 1079	. . 3 |- ( ( C e. (Moore ` X ) /\ U C_ V /\ V C_ X ) -> U C_ X )
10		mrcfval.f	. . . 4 |- F = (mrCls ` C )
11	10	mrcval 16270	. . 3 |- ( ( C e. (Moore ` X ) /\ U C_ X ) -> ( F ` U ) = |^| { s e. C | U C_ s } )
12	7, 9, 11	syl2anc 693	. 2 |- ( ( C e. (Moore ` X ) /\ U C_ V /\ V C_ X ) -> ( F ` U ) = |^| { s e. C | U C_ s } )
13	10	mrcval 16270	. . 3 |- ( ( C e. (Moore ` X ) /\ V C_ X ) -> ( F ` V ) = |^| { s e. C | V C_ s } )
14	13	3adant2 1080	. 2 |- ( ( C e. (Moore ` X ) /\ U C_ V /\ V C_ X ) -> ( F ` V ) = |^| { s e. C | V C_ s } )
15	6, 12, 14	3sstr4d 3648	1 |- ( ( C e. (Moore ` X ) /\ U C_ V /\ V C_ X ) -> ( F ` U ) C_ ( F ` V ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   |^|cint 4475   ` cfv 5888  Moorecmre 16242  mrClscmrc 16243

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-int 4476  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-mre 16246  df-mrc 16247

This theorem is referenced by:  mrcsscl  16280  mrcuni  16281  mrcssd  16284  ismrc  37264  isnacs3  37273