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Theorem mclsax 31201
Description: The closure is closed under axiom application. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mclsval.d 𝐷 = (mDV‘𝑇)
mclsval.e 𝐸 = (mEx‘𝑇)
mclsval.c 𝐶 = (mCls‘𝑇)
mclsval.1 (𝜑𝑇 ∈ mFS)
mclsval.2 (𝜑𝐾𝐷)
mclsval.3 (𝜑𝐵𝐸)
mclsax.a 𝐴 = (mAx‘𝑇)
mclsax.l 𝐿 = (mSubst‘𝑇)
mclsax.v 𝑉 = (mVR‘𝑇)
mclsax.h 𝐻 = (mVH‘𝑇)
mclsax.w 𝑊 = (mVars‘𝑇)
mclsax.4 (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴)
mclsax.5 (𝜑𝑆 ∈ ran 𝐿)
mclsax.6 ((𝜑𝑥𝑂) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
mclsax.7 ((𝜑𝑣𝑉) → (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))
mclsax.8 ((𝜑 ∧ (𝑥𝑀𝑦𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦))))) → 𝑎𝐾𝑏)
Assertion
Ref Expression
mclsax (𝜑 → (𝑆𝑃) ∈ (𝐾𝐶𝐵))
Distinct variable groups:   𝑣,𝐸   𝑎,𝑏,𝑣,𝑥,𝐻   𝑦,𝑣,𝐵,𝑥   𝑣,𝐶,𝑥   𝑥,𝐿,𝑦   𝑥,𝑂,𝑦   𝑦,𝑎,𝑆,𝑏,𝑣,𝑥   𝑀,𝑎,𝑏,𝑥,𝑦   𝑥,𝑃,𝑦   𝑥,𝑇,𝑦   𝜑,𝑎,𝑏,𝑣,𝑥,𝑦   𝑣,𝑉,𝑥   𝑊,𝑎,𝑏,𝑥   𝐾,𝑎,𝑏,𝑣,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑣,𝑎,𝑏)   𝐵(𝑎,𝑏)   𝐶(𝑦,𝑎,𝑏)   𝐷(𝑥,𝑦,𝑣,𝑎,𝑏)   𝑃(𝑣,𝑎,𝑏)   𝑇(𝑣,𝑎,𝑏)   𝐸(𝑥,𝑦,𝑎,𝑏)   𝐻(𝑦)   𝐿(𝑣,𝑎,𝑏)   𝑀(𝑣)   𝑂(𝑣,𝑎,𝑏)   𝑉(𝑦,𝑎,𝑏)   𝑊(𝑦,𝑣)

Proof of Theorem mclsax
Dummy variables 𝑐 𝑚 𝑜 𝑝 𝑠 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 abid 2609 . . . . . . . 8 (𝑐 ∈ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))} ↔ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐))))
2 intss1 4462 . . . . . . . 8 (𝑐 ∈ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))} → {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))} ⊆ 𝑐)
31, 2sylbir 225 . . . . . . 7 (((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐))) → {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))} ⊆ 𝑐)
4 mclsval.d . . . . . . . . 9 𝐷 = (mDV‘𝑇)
5 mclsval.e . . . . . . . . 9 𝐸 = (mEx‘𝑇)
6 mclsval.c . . . . . . . . 9 𝐶 = (mCls‘𝑇)
7 mclsval.1 . . . . . . . . 9 (𝜑𝑇 ∈ mFS)
8 mclsval.2 . . . . . . . . 9 (𝜑𝐾𝐷)
9 mclsval.3 . . . . . . . . 9 (𝜑𝐵𝐸)
10 mclsax.h . . . . . . . . 9 𝐻 = (mVH‘𝑇)
11 mclsax.a . . . . . . . . 9 𝐴 = (mAx‘𝑇)
12 mclsax.l . . . . . . . . 9 𝐿 = (mSubst‘𝑇)
13 mclsax.w . . . . . . . . 9 𝑊 = (mVars‘𝑇)
144, 5, 6, 7, 8, 9, 10, 11, 12, 13mclsval 31195 . . . . . . . 8 (𝜑 → (𝐾𝐶𝐵) = {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))})
1514sseq1d 3616 . . . . . . 7 (𝜑 → ((𝐾𝐶𝐵) ⊆ 𝑐 {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))} ⊆ 𝑐))
163, 15syl5ibr 236 . . . . . 6 (𝜑 → (((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐))) → (𝐾𝐶𝐵) ⊆ 𝑐))
17 sstr2 3594 . . . . . . . . . . . . . . 15 ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) → ((𝐾𝐶𝐵) ⊆ 𝑐 → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐))
1817com12 32 . . . . . . . . . . . . . 14 ((𝐾𝐶𝐵) ⊆ 𝑐 → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐))
1918anim1d 587 . . . . . . . . . . . . 13 ((𝐾𝐶𝐵) ⊆ 𝑐 → (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾))))
2019imim1d 82 . . . . . . . . . . . 12 ((𝐾𝐶𝐵) ⊆ 𝑐 → ((((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐) → (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))
2120ralimdv 2958 . . . . . . . . . . 11 ((𝐾𝐶𝐵) ⊆ 𝑐 → (∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐) → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))
2221imim2d 57 . . . . . . . . . 10 ((𝐾𝐶𝐵) ⊆ 𝑐 → ((⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)) → (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐))))
2322alimdv 1842 . . . . . . . . 9 ((𝐾𝐶𝐵) ⊆ 𝑐 → (∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)) → ∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐))))
24232alimdv 1844 . . . . . . . 8 ((𝐾𝐶𝐵) ⊆ 𝑐 → (∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)) → ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐))))
2524com12 32 . . . . . . 7 (∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)) → ((𝐾𝐶𝐵) ⊆ 𝑐 → ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐))))
2625adantl 482 . . . . . 6 (((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐))) → ((𝐾𝐶𝐵) ⊆ 𝑐 → ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐))))
2716, 26sylcom 30 . . . . 5 (𝜑 → (((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐))) → ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐))))
28 eqid 2621 . . . . . . . 8 (mPreSt‘𝑇) = (mPreSt‘𝑇)
29 eqid 2621 . . . . . . . 8 (mStat‘𝑇) = (mStat‘𝑇)
3028, 29mstapst 31179 . . . . . . 7 (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
3111, 29maxsta 31186 . . . . . . . . 9 (𝑇 ∈ mFS → 𝐴 ⊆ (mStat‘𝑇))
327, 31syl 17 . . . . . . . 8 (𝜑𝐴 ⊆ (mStat‘𝑇))
33 mclsax.4 . . . . . . . 8 (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴)
3432, 33sseldd 3588 . . . . . . 7 (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mStat‘𝑇))
3530, 34sseldi 3585 . . . . . 6 (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇))
3628mpstrcl 31173 . . . . . 6 (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) → (𝑀 ∈ V ∧ 𝑂 ∈ V ∧ 𝑃 ∈ V))
37 simp1 1059 . . . . . . . . . 10 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → 𝑚 = 𝑀)
38 simp2 1060 . . . . . . . . . 10 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → 𝑜 = 𝑂)
39 simp3 1061 . . . . . . . . . 10 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → 𝑝 = 𝑃)
4037, 38, 39oteq123d 4390 . . . . . . . . 9 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → ⟨𝑚, 𝑜, 𝑝⟩ = ⟨𝑀, 𝑂, 𝑃⟩)
4140eleq1d 2683 . . . . . . . 8 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ↔ ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴))
4238uneq1d 3749 . . . . . . . . . . . . 13 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (𝑜 ∪ ran 𝐻) = (𝑂 ∪ ran 𝐻))
4342imaeq2d 5430 . . . . . . . . . . . 12 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (𝑠 “ (𝑜 ∪ ran 𝐻)) = (𝑠 “ (𝑂 ∪ ran 𝐻)))
4443sseq1d 3616 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ (𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵)))
4537breqd 4629 . . . . . . . . . . . . 13 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (𝑥𝑚𝑦𝑥𝑀𝑦))
4645imbi1d 331 . . . . . . . . . . . 12 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → ((𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾) ↔ (𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)))
47462albidv 1848 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾) ↔ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)))
4844, 47anbi12d 746 . . . . . . . . . 10 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) ↔ ((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾))))
4939fveq2d 6157 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (𝑠𝑝) = (𝑠𝑃))
5049eleq1d 2683 . . . . . . . . . 10 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → ((𝑠𝑝) ∈ 𝑐 ↔ (𝑠𝑃) ∈ 𝑐))
5148, 50imbi12d 334 . . . . . . . . 9 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → ((((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐) ↔ (((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑃) ∈ 𝑐)))
5251ralbidv 2981 . . . . . . . 8 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐) ↔ ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑃) ∈ 𝑐)))
5341, 52imbi12d 334 . . . . . . 7 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → ((⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)) ↔ (⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑃) ∈ 𝑐))))
5453spc3gv 3287 . . . . . 6 ((𝑀 ∈ V ∧ 𝑂 ∈ V ∧ 𝑃 ∈ V) → (∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)) → (⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑃) ∈ 𝑐))))
5535, 36, 543syl 18 . . . . 5 (𝜑 → (∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)) → (⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑃) ∈ 𝑐))))
56 elun 3736 . . . . . . . . . . 11 (𝑥 ∈ (𝑂 ∪ ran 𝐻) ↔ (𝑥𝑂𝑥 ∈ ran 𝐻))
57 mclsax.6 . . . . . . . . . . . 12 ((𝜑𝑥𝑂) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
58 mclsax.7 . . . . . . . . . . . . . . 15 ((𝜑𝑣𝑉) → (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))
5958ralrimiva 2961 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑣𝑉 (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))
60 mclsax.v . . . . . . . . . . . . . . . . 17 𝑉 = (mVR‘𝑇)
6160, 5, 10mvhf 31190 . . . . . . . . . . . . . . . 16 (𝑇 ∈ mFS → 𝐻:𝑉𝐸)
627, 61syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐻:𝑉𝐸)
63 ffn 6007 . . . . . . . . . . . . . . 15 (𝐻:𝑉𝐸𝐻 Fn 𝑉)
64 fveq2 6153 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝐻𝑣) → (𝑆𝑥) = (𝑆‘(𝐻𝑣)))
6564eleq1d 2683 . . . . . . . . . . . . . . . 16 (𝑥 = (𝐻𝑣) → ((𝑆𝑥) ∈ (𝐾𝐶𝐵) ↔ (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵)))
6665ralrn 6323 . . . . . . . . . . . . . . 15 (𝐻 Fn 𝑉 → (∀𝑥 ∈ ran 𝐻(𝑆𝑥) ∈ (𝐾𝐶𝐵) ↔ ∀𝑣𝑉 (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵)))
6762, 63, 663syl 18 . . . . . . . . . . . . . 14 (𝜑 → (∀𝑥 ∈ ran 𝐻(𝑆𝑥) ∈ (𝐾𝐶𝐵) ↔ ∀𝑣𝑉 (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵)))
6859, 67mpbird 247 . . . . . . . . . . . . 13 (𝜑 → ∀𝑥 ∈ ran 𝐻(𝑆𝑥) ∈ (𝐾𝐶𝐵))
6968r19.21bi 2927 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ran 𝐻) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
7057, 69jaodan 825 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑂𝑥 ∈ ran 𝐻)) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
7156, 70sylan2b 492 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑂 ∪ ran 𝐻)) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
7271ralrimiva 2961 . . . . . . . . 9 (𝜑 → ∀𝑥 ∈ (𝑂 ∪ ran 𝐻)(𝑆𝑥) ∈ (𝐾𝐶𝐵))
73 mclsax.5 . . . . . . . . . . . 12 (𝜑𝑆 ∈ ran 𝐿)
7412, 5msubf 31164 . . . . . . . . . . . 12 (𝑆 ∈ ran 𝐿𝑆:𝐸𝐸)
7573, 74syl 17 . . . . . . . . . . 11 (𝜑𝑆:𝐸𝐸)
76 ffun 6010 . . . . . . . . . . 11 (𝑆:𝐸𝐸 → Fun 𝑆)
7775, 76syl 17 . . . . . . . . . 10 (𝜑 → Fun 𝑆)
784, 5, 28elmpst 31168 . . . . . . . . . . . . . . 15 (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑀𝐷𝑀 = 𝑀) ∧ (𝑂𝐸𝑂 ∈ Fin) ∧ 𝑃𝐸))
7935, 78sylib 208 . . . . . . . . . . . . . 14 (𝜑 → ((𝑀𝐷𝑀 = 𝑀) ∧ (𝑂𝐸𝑂 ∈ Fin) ∧ 𝑃𝐸))
8079simp2d 1072 . . . . . . . . . . . . 13 (𝜑 → (𝑂𝐸𝑂 ∈ Fin))
8180simpld 475 . . . . . . . . . . . 12 (𝜑𝑂𝐸)
82 fdm 6013 . . . . . . . . . . . . 13 (𝑆:𝐸𝐸 → dom 𝑆 = 𝐸)
8375, 82syl 17 . . . . . . . . . . . 12 (𝜑 → dom 𝑆 = 𝐸)
8481, 83sseqtr4d 3626 . . . . . . . . . . 11 (𝜑𝑂 ⊆ dom 𝑆)
85 frn 6015 . . . . . . . . . . . . 13 (𝐻:𝑉𝐸 → ran 𝐻𝐸)
8662, 85syl 17 . . . . . . . . . . . 12 (𝜑 → ran 𝐻𝐸)
8786, 83sseqtr4d 3626 . . . . . . . . . . 11 (𝜑 → ran 𝐻 ⊆ dom 𝑆)
8884, 87unssd 3772 . . . . . . . . . 10 (𝜑 → (𝑂 ∪ ran 𝐻) ⊆ dom 𝑆)
89 funimass4 6209 . . . . . . . . . 10 ((Fun 𝑆 ∧ (𝑂 ∪ ran 𝐻) ⊆ dom 𝑆) → ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ ∀𝑥 ∈ (𝑂 ∪ ran 𝐻)(𝑆𝑥) ∈ (𝐾𝐶𝐵)))
9077, 88, 89syl2anc 692 . . . . . . . . 9 (𝜑 → ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ ∀𝑥 ∈ (𝑂 ∪ ran 𝐻)(𝑆𝑥) ∈ (𝐾𝐶𝐵)))
9172, 90mpbird 247 . . . . . . . 8 (𝜑 → (𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵))
92 mclsax.8 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑀𝑦𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦))))) → 𝑎𝐾𝑏)
93923exp2 1282 . . . . . . . . . . . . 13 (𝜑 → (𝑥𝑀𝑦 → (𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) → (𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦))) → 𝑎𝐾𝑏))))
9493imp4b 612 . . . . . . . . . . . 12 ((𝜑𝑥𝑀𝑦) → ((𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦)))) → 𝑎𝐾𝑏))
9594ralrimivv 2965 . . . . . . . . . . 11 ((𝜑𝑥𝑀𝑦) → ∀𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥)))∀𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦)))𝑎𝐾𝑏)
96 dfss3 3577 . . . . . . . . . . . 12 (((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾 ↔ ∀𝑧 ∈ ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦))))𝑧𝐾)
97 eleq1 2686 . . . . . . . . . . . . . 14 (𝑧 = ⟨𝑎, 𝑏⟩ → (𝑧𝐾 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝐾))
98 df-br 4619 . . . . . . . . . . . . . 14 (𝑎𝐾𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝐾)
9997, 98syl6bbr 278 . . . . . . . . . . . . 13 (𝑧 = ⟨𝑎, 𝑏⟩ → (𝑧𝐾𝑎𝐾𝑏))
10099ralxp 5228 . . . . . . . . . . . 12 (∀𝑧 ∈ ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦))))𝑧𝐾 ↔ ∀𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥)))∀𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦)))𝑎𝐾𝑏)
10196, 100bitri 264 . . . . . . . . . . 11 (((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾 ↔ ∀𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥)))∀𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦)))𝑎𝐾𝑏)
10295, 101sylibr 224 . . . . . . . . . 10 ((𝜑𝑥𝑀𝑦) → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾)
103102ex 450 . . . . . . . . 9 (𝜑 → (𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾))
104103alrimivv 1853 . . . . . . . 8 (𝜑 → ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾))
10591, 104jca 554 . . . . . . 7 (𝜑 → ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾)))
106 imaeq1 5425 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (𝑠 “ (𝑂 ∪ ran 𝐻)) = (𝑆 “ (𝑂 ∪ ran 𝐻)))
107106sseq1d 3616 . . . . . . . . . . 11 (𝑠 = 𝑆 → ((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ (𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵)))
108 fveq1 6152 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑆 → (𝑠‘(𝐻𝑥)) = (𝑆‘(𝐻𝑥)))
109108fveq2d 6157 . . . . . . . . . . . . . . 15 (𝑠 = 𝑆 → (𝑊‘(𝑠‘(𝐻𝑥))) = (𝑊‘(𝑆‘(𝐻𝑥))))
110 fveq1 6152 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑆 → (𝑠‘(𝐻𝑦)) = (𝑆‘(𝐻𝑦)))
111110fveq2d 6157 . . . . . . . . . . . . . . 15 (𝑠 = 𝑆 → (𝑊‘(𝑠‘(𝐻𝑦))) = (𝑊‘(𝑆‘(𝐻𝑦))))
112109, 111xpeq12d 5105 . . . . . . . . . . . . . 14 (𝑠 = 𝑆 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) = ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))))
113112sseq1d 3616 . . . . . . . . . . . . 13 (𝑠 = 𝑆 → (((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾 ↔ ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾))
114113imbi2d 330 . . . . . . . . . . . 12 (𝑠 = 𝑆 → ((𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾) ↔ (𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾)))
1151142albidv 1848 . . . . . . . . . . 11 (𝑠 = 𝑆 → (∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾) ↔ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾)))
116107, 115anbi12d 746 . . . . . . . . . 10 (𝑠 = 𝑆 → (((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) ↔ ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾))))
117 fveq1 6152 . . . . . . . . . . 11 (𝑠 = 𝑆 → (𝑠𝑃) = (𝑆𝑃))
118117eleq1d 2683 . . . . . . . . . 10 (𝑠 = 𝑆 → ((𝑠𝑃) ∈ 𝑐 ↔ (𝑆𝑃) ∈ 𝑐))
119116, 118imbi12d 334 . . . . . . . . 9 (𝑠 = 𝑆 → ((((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑃) ∈ 𝑐) ↔ (((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑆𝑃) ∈ 𝑐)))
120119rspcv 3294 . . . . . . . 8 (𝑆 ∈ ran 𝐿 → (∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑃) ∈ 𝑐) → (((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑆𝑃) ∈ 𝑐)))
12173, 120syl 17 . . . . . . 7 (𝜑 → (∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑃) ∈ 𝑐) → (((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑆𝑃) ∈ 𝑐)))
122105, 121mpid 44 . . . . . 6 (𝜑 → (∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑃) ∈ 𝑐) → (𝑆𝑃) ∈ 𝑐))
12333, 122embantd 59 . . . . 5 (𝜑 → ((⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑃) ∈ 𝑐)) → (𝑆𝑃) ∈ 𝑐))
12427, 55, 1233syld 60 . . . 4 (𝜑 → (((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐))) → (𝑆𝑃) ∈ 𝑐))
125124alrimiv 1852 . . 3 (𝜑 → ∀𝑐(((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐))) → (𝑆𝑃) ∈ 𝑐))
126 fvex 6163 . . . 4 (𝑆𝑃) ∈ V
127126elintab 4457 . . 3 ((𝑆𝑃) ∈ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))} ↔ ∀𝑐(((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐))) → (𝑆𝑃) ∈ 𝑐))
128125, 127sylibr 224 . 2 (𝜑 → (𝑆𝑃) ∈ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))})
129128, 14eleqtrrd 2701 1 (𝜑 → (𝑆𝑃) ∈ (𝐾𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1987  {cab 2607  wral 2907  Vcvv 3189  cun 3557  wss 3559  cop 4159  cotp 4161   cint 4445   class class class wbr 4618   × cxp 5077  ccnv 5078  dom cdm 5079  ran crn 5080  cima 5082  Fun wfun 5846   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  Fincfn 7906  mVRcmvar 31093  mAxcmax 31097  mExcmex 31099  mDVcmdv 31100  mVarscmvrs 31101  mSubstcmsub 31103  mVHcmvh 31104  mPreStcmpst 31105  mStatcmsta 31107  mFScmfs 31108  mClscmcls 31109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-ot 4162  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-card 8716  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-2 11030  df-n0 11244  df-z 11329  df-uz 11639  df-fz 12276  df-fzo 12414  df-seq 12749  df-hash 13065  df-word 13245  df-concat 13247  df-s1 13248  df-struct 15790  df-ndx 15791  df-slot 15792  df-base 15793  df-sets 15794  df-ress 15795  df-plusg 15882  df-0g 16030  df-gsum 16031  df-mgm 17170  df-sgrp 17212  df-mnd 17223  df-submnd 17264  df-frmd 17314  df-mrex 31118  df-mex 31119  df-mrsub 31122  df-msub 31123  df-mvh 31124  df-mpst 31125  df-msr 31126  df-msta 31127  df-mfs 31128  df-mcls 31129
This theorem is referenced by:  mclsppslem  31215
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