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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssmcls | Structured version Visualization version GIF version |
Description: The original expressions are also in the closure. (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 |
---|---|
ssmcls | ⊢ (𝜑 → 𝐵 ⊆ (𝐾𝐶𝐵)) |
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 2779 | . . 3 ⊢ (mVH‘𝑇) = (mVH‘𝑇) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ssmclslem 32329 | . 2 ⊢ (𝜑 → (𝐵 ∪ ran (mVH‘𝑇)) ⊆ (𝐾𝐶𝐵)) |
9 | 8 | unssad 4052 | 1 ⊢ (𝜑 → 𝐵 ⊆ (𝐾𝐶𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 ⊆ wss 3830 ran crn 5408 ‘cfv 6188 (class class class)co 6976 mExcmex 32231 mDVcmdv 32232 mVHcmvh 32236 mFScmfs 32240 mClscmcls 32241 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-ot 4450 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-map 8208 df-pm 8209 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-card 9162 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-n0 11708 df-z 11794 df-uz 12059 df-fz 12709 df-fzo 12850 df-seq 13185 df-hash 13506 df-word 13673 df-concat 13734 df-s1 13759 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-0g 16571 df-gsum 16572 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-submnd 17804 df-frmd 17855 df-mrex 32250 df-mex 32251 df-mrsub 32254 df-msub 32255 df-mvh 32256 df-mpst 32257 df-msr 32258 df-msta 32259 df-mfs 32260 df-mcls 32261 |
This theorem is referenced by: (None) |
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