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Theorem ss2mcls 35086
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 4174 . . . . . 6 (π‘Œ βŠ† 𝐡 β†’ (π‘Œ βˆͺ ran (mVHβ€˜π‘‡)) βŠ† (𝐡 βˆͺ ran (mVHβ€˜π‘‡)))
3 sstr2 3984 . . . . . 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 3984 . . . . . . . . . . . . . 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 611 . . . . . . . . . 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 3163 . . . . . . . 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 608 . . . 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 4055 . . 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 4966 . . 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 3987 . . 3 (πœ‘ β†’ 𝑋 βŠ† 𝐷)
26 mclsval.3 . . . 4 (πœ‘ β†’ 𝐡 βŠ† 𝐸)
271, 26sstrd 3987 . . 3 (πœ‘ β†’ π‘Œ βŠ† 𝐸)
28 eqid 2726 . . 3 (mVHβ€˜π‘‡) = (mVHβ€˜π‘‡)
29 eqid 2726 . . 3 (mAxβ€˜π‘‡) = (mAxβ€˜π‘‡)
30 eqid 2726 . . 3 (mSubstβ€˜π‘‡) = (mSubstβ€˜π‘‡)
31 eqid 2726 . . 3 (mVarsβ€˜π‘‡) = (mVarsβ€˜π‘‡)
3220, 21, 22, 23, 25, 27, 28, 29, 30, 31mclsval 35081 . 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 35081 . 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 4024 1 (πœ‘ β†’ (π‘‹πΆπ‘Œ) βŠ† (𝐾𝐢𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395  βˆ€wal 1531   = wceq 1533   ∈ wcel 2098  {cab 2703  βˆ€wral 3055   βˆͺ cun 3941   βŠ† wss 3943  βŸ¨cotp 4631  βˆ© cint 4943   class class class wbr 5141   Γ— cxp 5667  ran crn 5670   β€œ cima 5672  β€˜cfv 6536  (class class class)co 7404  mAxcmax 34983  mExcmex 34985  mDVcmdv 34986  mVarscmvrs 34987  mSubstcmsub 34989  mVHcmvh 34990  mFScmfs 34994  mClscmcls 34995
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-ot 4632  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-er 8702  df-map 8821  df-pm 8822  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-card 9933  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-n0 12474  df-z 12560  df-uz 12824  df-fz 13488  df-fzo 13631  df-seq 13970  df-hash 14293  df-word 14468  df-concat 14524  df-s1 14549  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17151  df-ress 17180  df-plusg 17216  df-0g 17393  df-gsum 17394  df-mgm 18570  df-sgrp 18649  df-mnd 18665  df-submnd 18711  df-frmd 18771  df-mrex 35004  df-mex 35005  df-mrsub 35008  df-msub 35009  df-mvh 35010  df-mpst 35011  df-msr 35012  df-msta 35013  df-mfs 35014  df-mcls 35015
This theorem is referenced by:  mthmpps  35100
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