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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssmclslem | Structured version Visualization version GIF version |
Description: Lemma for ssmcls 32334. (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) |
ssmclslem.h | ⊢ 𝐻 = (mVH‘𝑇) |
Ref | Expression |
---|---|
ssmclslem | ⊢ (𝜑 → (𝐵 ∪ ran 𝐻) ⊆ (𝐾𝐶𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 475 | . . . . 5 ⊢ (((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘(𝐻‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) → (𝐵 ∪ ran 𝐻) ⊆ 𝑐) | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → (((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘(𝐻‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) → (𝐵 ∪ ran 𝐻) ⊆ 𝑐)) |
3 | 2 | alrimiv 1886 | . . 3 ⊢ (𝜑 → ∀𝑐(((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘(𝐻‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) → (𝐵 ∪ ran 𝐻) ⊆ 𝑐)) |
4 | ssintab 4760 | . . 3 ⊢ ((𝐵 ∪ ran 𝐻) ⊆ ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘(𝐻‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ↔ ∀𝑐(((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘(𝐻‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) → (𝐵 ∪ ran 𝐻) ⊆ 𝑐)) | |
5 | 3, 4 | sylibr 226 | . 2 ⊢ (𝜑 → (𝐵 ∪ ran 𝐻) ⊆ ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘(𝐻‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))}) |
6 | mclsval.d | . . 3 ⊢ 𝐷 = (mDV‘𝑇) | |
7 | mclsval.e | . . 3 ⊢ 𝐸 = (mEx‘𝑇) | |
8 | mclsval.c | . . 3 ⊢ 𝐶 = (mCls‘𝑇) | |
9 | mclsval.1 | . . 3 ⊢ (𝜑 → 𝑇 ∈ mFS) | |
10 | mclsval.2 | . . 3 ⊢ (𝜑 → 𝐾 ⊆ 𝐷) | |
11 | mclsval.3 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐸) | |
12 | ssmclslem.h | . . 3 ⊢ 𝐻 = (mVH‘𝑇) | |
13 | eqid 2772 | . . 3 ⊢ (mAx‘𝑇) = (mAx‘𝑇) | |
14 | eqid 2772 | . . 3 ⊢ (mSubst‘𝑇) = (mSubst‘𝑇) | |
15 | eqid 2772 | . . 3 ⊢ (mVars‘𝑇) = (mVars‘𝑇) | |
16 | 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 | mclsval 32330 | . 2 ⊢ (𝜑 → (𝐾𝐶𝐵) = ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘(𝐻‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))}) |
17 | 5, 16 | sseqtr4d 3892 | 1 ⊢ (𝜑 → (𝐵 ∪ ran 𝐻) ⊆ (𝐾𝐶𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∀wal 1505 = wceq 1507 ∈ wcel 2050 {cab 2752 ∀wral 3082 ∪ cun 3821 ⊆ wss 3823 〈cotp 4443 ∩ cint 4743 class class class wbr 4923 × cxp 5399 ran crn 5402 “ cima 5404 ‘cfv 6182 (class class class)co 6970 mAxcmax 32232 mExcmex 32234 mDVcmdv 32235 mVarscmvrs 32236 mSubstcmsub 32238 mVHcmvh 32239 mFScmfs 32243 mClscmcls 32244 |
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 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 |
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 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-ot 4444 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7495 df-2nd 7496 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-oadd 7903 df-er 8083 df-map 8202 df-pm 8203 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-card 9156 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-nn 11434 df-2 11497 df-n0 11702 df-z 11788 df-uz 12053 df-fz 12703 df-fzo 12844 df-seq 13179 df-hash 13500 df-word 13667 df-concat 13728 df-s1 13753 df-struct 16335 df-ndx 16336 df-slot 16337 df-base 16339 df-sets 16340 df-ress 16341 df-plusg 16428 df-0g 16565 df-gsum 16566 df-mgm 17704 df-sgrp 17746 df-mnd 17757 df-submnd 17798 df-frmd 17849 df-mrex 32253 df-mex 32254 df-mrsub 32257 df-msub 32258 df-mvh 32259 df-mpst 32260 df-msr 32261 df-msta 32262 df-mfs 32263 df-mcls 32264 |
This theorem is referenced by: vhmcls 32333 ssmcls 32334 |
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