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Mirrors > Home > MPE Home > Th. List > Mathboxes > mclsssv | Structured version Visualization version GIF version |
Description: The closure of a set of expressions is a set of expressions. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mclsval.d | ⊢ 𝐷 = (mDV‘𝑇) |
mclsval.e | ⊢ 𝐸 = (mEx‘𝑇) |
mclsval.c | ⊢ 𝐶 = (mCls‘𝑇) |
mclsval.1 | ⊢ (𝜑 → 𝑇 ∈ mFS) |
mclsval.2 | ⊢ (𝜑 → 𝐾 ⊆ 𝐷) |
mclsval.3 | ⊢ (𝜑 → 𝐵 ⊆ 𝐸) |
Ref | Expression |
---|---|
mclsssv | ⊢ (𝜑 → (𝐾𝐶𝐵) ⊆ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mclsval.d | . . 3 ⊢ 𝐷 = (mDV‘𝑇) | |
2 | mclsval.e | . . 3 ⊢ 𝐸 = (mEx‘𝑇) | |
3 | mclsval.c | . . 3 ⊢ 𝐶 = (mCls‘𝑇) | |
4 | mclsval.1 | . . 3 ⊢ (𝜑 → 𝑇 ∈ mFS) | |
5 | mclsval.2 | . . 3 ⊢ (𝜑 → 𝐾 ⊆ 𝐷) | |
6 | mclsval.3 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐸) | |
7 | eqid 2622 | . . 3 ⊢ (mVH‘𝑇) = (mVH‘𝑇) | |
8 | eqid 2622 | . . 3 ⊢ (mAx‘𝑇) = (mAx‘𝑇) | |
9 | eqid 2622 | . . 3 ⊢ (mSubst‘𝑇) = (mSubst‘𝑇) | |
10 | eqid 2622 | . . 3 ⊢ (mVars‘𝑇) = (mVars‘𝑇) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | mclsval 31460 | . 2 ⊢ (𝜑 → (𝐾𝐶𝐵) = ∩ {𝑐 ∣ ((𝐵 ∪ ran (mVH‘𝑇)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))}) |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | mclsssvlem 31459 | . 2 ⊢ (𝜑 → ∩ {𝑐 ∣ ((𝐵 ∪ ran (mVH‘𝑇)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ 𝐸) |
13 | 11, 12 | eqsstrd 3639 | 1 ⊢ (𝜑 → (𝐾𝐶𝐵) ⊆ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∀wal 1481 = wceq 1483 ∈ wcel 1990 {cab 2608 ∀wral 2912 ∪ cun 3572 ⊆ wss 3574 〈cotp 4185 ∩ cint 4475 class class class wbr 4653 × cxp 5112 ran crn 5115 “ cima 5117 ‘cfv 5888 (class class class)co 6650 mAxcmax 31362 mExcmex 31364 mDVcmdv 31365 mVarscmvrs 31366 mSubstcmsub 31368 mVHcmvh 31369 mFScmfs 31373 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 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 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-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 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-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-gsum 16103 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-frmd 17386 df-mrex 31383 df-mex 31384 df-mrsub 31387 df-msub 31388 df-mvh 31389 df-mpst 31390 df-msr 31391 df-msta 31392 df-mfs 31393 df-mcls 31394 |
This theorem is referenced by: (None) |
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