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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssmclslem | Structured version Visualization version GIF version |
Description: Lemma for ssmcls 32814. (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 485 | . . . . 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 1928 | . . 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 4893 | . . 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 236 | . 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 2821 | . . 3 ⊢ (mAx‘𝑇) = (mAx‘𝑇) | |
14 | eqid 2821 | . . 3 ⊢ (mSubst‘𝑇) = (mSubst‘𝑇) | |
15 | eqid 2821 | . . 3 ⊢ (mVars‘𝑇) = (mVars‘𝑇) | |
16 | 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 | mclsval 32810 | . 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 4008 | 1 ⊢ (𝜑 → (𝐵 ∪ ran 𝐻) ⊆ (𝐾𝐶𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1535 = wceq 1537 ∈ wcel 2114 {cab 2799 ∀wral 3138 ∪ cun 3934 ⊆ wss 3936 〈cotp 4575 ∩ cint 4876 class class class wbr 5066 × cxp 5553 ran crn 5556 “ cima 5558 ‘cfv 6355 (class class class)co 7156 mAxcmax 32712 mExcmex 32714 mDVcmdv 32715 mVarscmvrs 32716 mSubstcmsub 32718 mVHcmvh 32719 mFScmfs 32723 mClscmcls 32724 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-0g 16715 df-gsum 16716 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-frmd 18014 df-mrex 32733 df-mex 32734 df-mrsub 32737 df-msub 32738 df-mvh 32739 df-mpst 32740 df-msr 32741 df-msta 32742 df-mfs 32743 df-mcls 32744 |
This theorem is referenced by: vhmcls 32813 ssmcls 32814 |
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