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Theorem ss2mcls 35211
Description: The closure is monotonic under subsets of the original set of expressions and the set of disjoint variable conditions. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mclsval.d 𝐷 = (mDVβ€˜π‘‡)
mclsval.e 𝐸 = (mExβ€˜π‘‡)
mclsval.c 𝐢 = (mClsβ€˜π‘‡)
mclsval.1 (πœ‘ β†’ 𝑇 ∈ mFS)
mclsval.2 (πœ‘ β†’ 𝐾 βŠ† 𝐷)
mclsval.3 (πœ‘ β†’ 𝐡 βŠ† 𝐸)
ss2mcls.4 (πœ‘ β†’ 𝑋 βŠ† 𝐾)
ss2mcls.5 (πœ‘ β†’ π‘Œ βŠ† 𝐡)
Assertion
Ref Expression
ss2mcls (πœ‘ β†’ (π‘‹πΆπ‘Œ) βŠ† (𝐾𝐢𝐡))

Proof of Theorem ss2mcls
Dummy variables 𝑐 π‘š π‘œ 𝑝 𝑠 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ss2mcls.5 . . . . . 6 (πœ‘ β†’ π‘Œ βŠ† 𝐡)
2 unss1 4181 . . . . . 6 (π‘Œ βŠ† 𝐡 β†’ (π‘Œ βˆͺ ran (mVHβ€˜π‘‡)) βŠ† (𝐡 βˆͺ ran (mVHβ€˜π‘‡)))
3 sstr2 3989 . . . . . 6 ((π‘Œ βˆͺ ran (mVHβ€˜π‘‡)) βŠ† (𝐡 βˆͺ ran (mVHβ€˜π‘‡)) β†’ ((𝐡 βˆͺ ran (mVHβ€˜π‘‡)) βŠ† 𝑐 β†’ (π‘Œ βˆͺ ran (mVHβ€˜π‘‡)) βŠ† 𝑐))
41, 2, 33syl 18 . . . . 5 (πœ‘ β†’ ((𝐡 βˆͺ ran (mVHβ€˜π‘‡)) βŠ† 𝑐 β†’ (π‘Œ βˆͺ ran (mVHβ€˜π‘‡)) βŠ† 𝑐))
5 ss2mcls.4 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝑋 βŠ† 𝐾)
6 sstr2 3989 . . . . . . . . . . . . . 14 ((((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝑋 β†’ (𝑋 βŠ† 𝐾 β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝐾))
75, 6syl5com 31 . . . . . . . . . . . . 13 (πœ‘ β†’ ((((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝑋 β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝐾))
87imim2d 57 . . . . . . . . . . . 12 (πœ‘ β†’ ((π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝑋) β†’ (π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝐾)))
982alimdv 1913 . . . . . . . . . . 11 (πœ‘ β†’ (βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝑋) β†’ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝐾)))
109anim2d 610 . . . . . . . . . 10 (πœ‘ β†’ (((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝑋)) β†’ ((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝐾))))
1110imim1d 82 . . . . . . . . 9 (πœ‘ β†’ ((((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝐾)) β†’ (π‘ β€˜π‘) ∈ 𝑐) β†’ (((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝑋)) β†’ (π‘ β€˜π‘) ∈ 𝑐)))
1211ralimdv 3166 . . . . . . . 8 (πœ‘ β†’ (βˆ€π‘  ∈ ran (mSubstβ€˜π‘‡)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝐾)) β†’ (π‘ β€˜π‘) ∈ 𝑐) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‡)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝑋)) β†’ (π‘ β€˜π‘) ∈ 𝑐)))
1312imim2d 57 . . . . . . 7 (πœ‘ β†’ ((βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‡) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‡)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝐾)) β†’ (π‘ β€˜π‘) ∈ 𝑐)) β†’ (βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‡) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‡)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝑋)) β†’ (π‘ β€˜π‘) ∈ 𝑐))))
1413alimdv 1911 . . . . . 6 (πœ‘ β†’ (βˆ€π‘(βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‡) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‡)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝐾)) β†’ (π‘ β€˜π‘) ∈ 𝑐)) β†’ βˆ€π‘(βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‡) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‡)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝑋)) β†’ (π‘ β€˜π‘) ∈ 𝑐))))
15142alimdv 1913 . . . . 5 (πœ‘ β†’ (βˆ€π‘šβˆ€π‘œβˆ€π‘(βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‡) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‡)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝐾)) β†’ (π‘ β€˜π‘) ∈ 𝑐)) β†’ βˆ€π‘šβˆ€π‘œβˆ€π‘(βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‡) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‡)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝑋)) β†’ (π‘ β€˜π‘) ∈ 𝑐))))
164, 15anim12d 607 . . . 4 (πœ‘ β†’ (((𝐡 βˆͺ ran (mVHβ€˜π‘‡)) βŠ† 𝑐 ∧ βˆ€π‘šβˆ€π‘œβˆ€π‘(βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‡) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‡)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝐾)) β†’ (π‘ β€˜π‘) ∈ 𝑐))) β†’ ((π‘Œ βˆͺ ran (mVHβ€˜π‘‡)) βŠ† 𝑐 ∧ βˆ€π‘šβˆ€π‘œβˆ€π‘(βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‡) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‡)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝑋)) β†’ (π‘ β€˜π‘) ∈ 𝑐)))))
1716ss2abdv 4060 . . 3 (πœ‘ β†’ {𝑐 ∣ ((𝐡 βˆͺ ran (mVHβ€˜π‘‡)) βŠ† 𝑐 ∧ βˆ€π‘šβˆ€π‘œβˆ€π‘(βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‡) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‡)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝐾)) β†’ (π‘ β€˜π‘) ∈ 𝑐)))} βŠ† {𝑐 ∣ ((π‘Œ βˆͺ ran (mVHβ€˜π‘‡)) βŠ† 𝑐 ∧ βˆ€π‘šβˆ€π‘œβˆ€π‘(βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‡) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‡)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝑋)) β†’ (π‘ β€˜π‘) ∈ 𝑐)))})
18 intss 4976 . . 3 ({𝑐 ∣ ((𝐡 βˆͺ ran (mVHβ€˜π‘‡)) βŠ† 𝑐 ∧ βˆ€π‘šβˆ€π‘œβˆ€π‘(βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‡) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‡)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝐾)) β†’ (π‘ β€˜π‘) ∈ 𝑐)))} βŠ† {𝑐 ∣ ((π‘Œ βˆͺ ran (mVHβ€˜π‘‡)) βŠ† 𝑐 ∧ βˆ€π‘šβˆ€π‘œβˆ€π‘(βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‡) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‡)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝑋)) β†’ (π‘ β€˜π‘) ∈ 𝑐)))} β†’ ∩ {𝑐 ∣ ((π‘Œ βˆͺ ran (mVHβ€˜π‘‡)) βŠ† 𝑐 ∧ βˆ€π‘šβˆ€π‘œβˆ€π‘(βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‡) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‡)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝑋)) β†’ (π‘ β€˜π‘) ∈ 𝑐)))} βŠ† ∩ {𝑐 ∣ ((𝐡 βˆͺ ran (mVHβ€˜π‘‡)) βŠ† 𝑐 ∧ βˆ€π‘šβˆ€π‘œβˆ€π‘(βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‡) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‡)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝐾)) β†’ (π‘ β€˜π‘) ∈ 𝑐)))})
1917, 18syl 17 . 2 (πœ‘ β†’ ∩ {𝑐 ∣ ((π‘Œ βˆͺ ran (mVHβ€˜π‘‡)) βŠ† 𝑐 ∧ βˆ€π‘šβˆ€π‘œβˆ€π‘(βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‡) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‡)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝑋)) β†’ (π‘ β€˜π‘) ∈ 𝑐)))} βŠ† ∩ {𝑐 ∣ ((𝐡 βˆͺ ran (mVHβ€˜π‘‡)) βŠ† 𝑐 ∧ βˆ€π‘šβˆ€π‘œβˆ€π‘(βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‡) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‡)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝐾)) β†’ (π‘ β€˜π‘) ∈ 𝑐)))})
20 mclsval.d . . 3 𝐷 = (mDVβ€˜π‘‡)
21 mclsval.e . . 3 𝐸 = (mExβ€˜π‘‡)
22 mclsval.c . . 3 𝐢 = (mClsβ€˜π‘‡)
23 mclsval.1 . . 3 (πœ‘ β†’ 𝑇 ∈ mFS)
24 mclsval.2 . . . 4 (πœ‘ β†’ 𝐾 βŠ† 𝐷)
255, 24sstrd 3992 . . 3 (πœ‘ β†’ 𝑋 βŠ† 𝐷)
26 mclsval.3 . . . 4 (πœ‘ β†’ 𝐡 βŠ† 𝐸)
271, 26sstrd 3992 . . 3 (πœ‘ β†’ π‘Œ βŠ† 𝐸)
28 eqid 2728 . . 3 (mVHβ€˜π‘‡) = (mVHβ€˜π‘‡)
29 eqid 2728 . . 3 (mAxβ€˜π‘‡) = (mAxβ€˜π‘‡)
30 eqid 2728 . . 3 (mSubstβ€˜π‘‡) = (mSubstβ€˜π‘‡)
31 eqid 2728 . . 3 (mVarsβ€˜π‘‡) = (mVarsβ€˜π‘‡)
3220, 21, 22, 23, 25, 27, 28, 29, 30, 31mclsval 35206 . 2 (πœ‘ β†’ (π‘‹πΆπ‘Œ) = ∩ {𝑐 ∣ ((π‘Œ βˆͺ ran (mVHβ€˜π‘‡)) βŠ† 𝑐 ∧ βˆ€π‘šβˆ€π‘œβˆ€π‘(βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‡) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‡)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝑋)) β†’ (π‘ β€˜π‘) ∈ 𝑐)))})
3320, 21, 22, 23, 24, 26, 28, 29, 30, 31mclsval 35206 . 2 (πœ‘ β†’ (𝐾𝐢𝐡) = ∩ {𝑐 ∣ ((𝐡 βˆͺ ran (mVHβ€˜π‘‡)) βŠ† 𝑐 ∧ βˆ€π‘šβˆ€π‘œβˆ€π‘(βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‡) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‡)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‡))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‡)β€˜(π‘ β€˜((mVHβ€˜π‘‡)β€˜π‘¦)))) βŠ† 𝐾)) β†’ (π‘ β€˜π‘) ∈ 𝑐)))})
3419, 32, 333sstr4d 4029 1 (πœ‘ β†’ (π‘‹πΆπ‘Œ) βŠ† (𝐾𝐢𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394  βˆ€wal 1531   = wceq 1533   ∈ wcel 2098  {cab 2705  βˆ€wral 3058   βˆͺ cun 3947   βŠ† wss 3949  βŸ¨cotp 4640  βˆ© cint 4953   class class class wbr 5152   Γ— cxp 5680  ran crn 5683   β€œ cima 5685  β€˜cfv 6553  (class class class)co 7426  mAxcmax 35108  mExcmex 35110  mDVcmdv 35111  mVarscmvrs 35112  mSubstcmsub 35114  mVHcmvh 35115  mFScmfs 35119  mClscmcls 35120
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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-ot 4641  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-pm 8854  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-n0 12511  df-z 12597  df-uz 12861  df-fz 13525  df-fzo 13668  df-seq 14007  df-hash 14330  df-word 14505  df-concat 14561  df-s1 14586  df-struct 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-0g 17430  df-gsum 17431  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-submnd 18748  df-frmd 18808  df-mrex 35129  df-mex 35130  df-mrsub 35133  df-msub 35134  df-mvh 35135  df-mpst 35136  df-msr 35137  df-msta 35138  df-mfs 35139  df-mcls 35140
This theorem is referenced by:  mthmpps  35225
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