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Theorem submrc 13854
 Description: In a closure system which is cut off above some level, closures below that level act as normal. (Contributed by Stefan O'Rear, 9-Mar-2015.)
Hypotheses
Ref Expression
submrc.f mrCls
submrc.g mrCls
Assertion
Ref Expression
submrc Moore

Proof of Theorem submrc
StepHypRef Expression
1 submre 13831 . . . 4 Moore Moore
213adant3 978 . . 3 Moore Moore
3 simp1 958 . . . 4 Moore Moore
4 submrc.f . . . 4 mrCls
5 simp3 960 . . . . 5 Moore
6 mress 13819 . . . . . 6 Moore
763adant3 978 . . . . 5 Moore
85, 7sstrd 3359 . . . 4 Moore
93, 4, 8mrcssidd 13851 . . 3 Moore
104mrccl 13837 . . . . 5 Moore
113, 8, 10syl2anc 644 . . . 4 Moore
124mrcsscl 13846 . . . . . 6 Moore
13123com23 1160 . . . . 5 Moore
14 fvex 5743 . . . . . 6
1514elpw 3806 . . . . 5
1613, 15sylibr 205 . . . 4 Moore
17 elin 3531 . . . 4
1811, 16, 17sylanbrc 647 . . 3 Moore
19 submrc.g . . . 4 mrCls
2019mrcsscl 13846 . . 3 Moore
212, 9, 18, 20syl3anc 1185 . 2 Moore
222, 19, 5mrcssidd 13851 . . 3 Moore
23 inss1 3562 . . . 4
2419mrccl 13837 . . . . 5 Moore
252, 5, 24syl2anc 644 . . . 4 Moore
2623, 25sseldi 3347 . . 3 Moore
274mrcsscl 13846 . . 3 Moore
283, 22, 26, 27syl3anc 1185 . 2 Moore
2921, 28eqssd 3366 1 Moore
 Colors of variables: wff set class Syntax hints:   wi 4   w3a 937   wceq 1653   wcel 1726   cin 3320   wss 3321  cpw 3800  cfv 5455  Moorecmre 13808  mrClscmrc 13809 This theorem is referenced by:  evlseu  19938 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463  df-mre 13812  df-mrc 13813
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