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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssmclslem | Structured version Visualization version GIF version |
Description: Lemma for ssmcls 35310. (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 481 | . . . . 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 1922 | . . 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 4969 | . . 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 233 | . 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 2725 | . . 3 ⊢ (mAx‘𝑇) = (mAx‘𝑇) | |
14 | eqid 2725 | . . 3 ⊢ (mSubst‘𝑇) = (mSubst‘𝑇) | |
15 | eqid 2725 | . . 3 ⊢ (mVars‘𝑇) = (mVars‘𝑇) | |
16 | 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 | mclsval 35306 | . 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 | sseqtrrd 4018 | 1 ⊢ (𝜑 → (𝐵 ∪ ran 𝐻) ⊆ (𝐾𝐶𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∀wal 1531 = wceq 1533 ∈ wcel 2098 {cab 2702 ∀wral 3050 ∪ cun 3942 ⊆ wss 3944 〈cotp 4638 ∩ cint 4950 class class class wbr 5149 × cxp 5676 ran crn 5679 “ cima 5681 ‘cfv 6549 (class class class)co 7419 mAxcmax 35208 mExcmex 35210 mDVcmdv 35211 mVarscmvrs 35212 mSubstcmsub 35214 mVHcmvh 35215 mFScmfs 35219 mClscmcls 35220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-ot 4639 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9969 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14008 df-hash 14331 df-word 14506 df-concat 14562 df-s1 14587 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17189 df-ress 17218 df-plusg 17254 df-0g 17431 df-gsum 17432 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18749 df-frmd 18814 df-mrex 35229 df-mex 35230 df-mrsub 35233 df-msub 35234 df-mvh 35235 df-mpst 35236 df-msr 35237 df-msta 35238 df-mfs 35239 df-mcls 35240 |
This theorem is referenced by: vhmcls 35309 ssmcls 35310 |
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