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Theorem mclsax 35554
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 2716 . . . . . . . 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 4968 . . . . . . . 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 235 . . . . . . 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 35548 . . . . . . . 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 4027 . . . . . . 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, 15imbitrrid 246 . . . . . 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 4002 . . . . . . . . . . . . . . 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 611 . . . . . . . . . . . . 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 3167 . . . . . . . . . . 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 1914 . . . . . . . . 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 1916 . . . . . . . 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 481 . . . . . 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 2735 . . . . . . . 8 (mPreSt‘𝑇) = (mPreSt‘𝑇)
29 eqid 2735 . . . . . . . 8 (mStat‘𝑇) = (mStat‘𝑇)
3028, 29mstapst 35532 . . . . . . 7 (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
3111, 29maxsta 35539 . . . . . . . . 9 (𝑇 ∈ mFS → 𝐴 ⊆ (mStat‘𝑇))
327, 31syl 17 . . . . . . . 8 (𝜑𝐴 ⊆ (mStat‘𝑇))
33 mclsax.4 . . . . . . . 8 (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴)
3432, 33sseldd 3996 . . . . . . 7 (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mStat‘𝑇))
3530, 34sselid 3993 . . . . . 6 (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇))
3628mpstrcl 35526 . . . . . 6 (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) → (𝑀 ∈ V ∧ 𝑂 ∈ V ∧ 𝑃 ∈ V))
37 simp1 1135 . . . . . . . . . 10 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → 𝑚 = 𝑀)
38 simp2 1136 . . . . . . . . . 10 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → 𝑜 = 𝑂)
39 simp3 1137 . . . . . . . . . 10 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → 𝑝 = 𝑃)
4037, 38, 39oteq123d 4893 . . . . . . . . 9 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → ⟨𝑚, 𝑜, 𝑝⟩ = ⟨𝑀, 𝑂, 𝑃⟩)
4140eleq1d 2824 . . . . . . . 8 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ↔ ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴))
4238uneq1d 4177 . . . . . . . . . . . . 13 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (𝑜 ∪ ran 𝐻) = (𝑂 ∪ ran 𝐻))
4342imaeq2d 6080 . . . . . . . . . . . 12 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (𝑠 “ (𝑜 ∪ ran 𝐻)) = (𝑠 “ (𝑂 ∪ ran 𝐻)))
4443sseq1d 4027 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ (𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵)))
4537breqd 5159 . . . . . . . . . . . . 13 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (𝑥𝑚𝑦𝑥𝑀𝑦))
4645imbi1d 341 . . . . . . . . . . . 12 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → ((𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾) ↔ (𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)))
47462albidv 1921 . . . . . . . . . . 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 632 . . . . . . . . . 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 6911 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (𝑠𝑝) = (𝑠𝑃))
5049eleq1d 2824 . . . . . . . . . 10 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → ((𝑠𝑝) ∈ 𝑐 ↔ (𝑠𝑃) ∈ 𝑐))
5148, 50imbi12d 344 . . . . . . . . 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 3176 . . . . . . . 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 344 . . . . . . 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 3604 . . . . . 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 4163 . . . . . . . . . . 11 (𝑥 ∈ (𝑂 ∪ ran 𝐻) ↔ (𝑥𝑂𝑥 ∈ ran 𝐻))
57 mclsax.6 . . . . . . . . . . . 12 ((𝜑𝑥𝑂) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
58 mclsax.7 . . . . . . . . . . . . . . 15 ((𝜑𝑣𝑉) → (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))
5958ralrimiva 3144 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑣𝑉 (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))
60 mclsax.v . . . . . . . . . . . . . . . . 17 𝑉 = (mVR‘𝑇)
6160, 5, 10mvhf 35543 . . . . . . . . . . . . . . . 16 (𝑇 ∈ mFS → 𝐻:𝑉𝐸)
627, 61syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐻:𝑉𝐸)
63 ffn 6737 . . . . . . . . . . . . . . 15 (𝐻:𝑉𝐸𝐻 Fn 𝑉)
64 fveq2 6907 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝐻𝑣) → (𝑆𝑥) = (𝑆‘(𝐻𝑣)))
6564eleq1d 2824 . . . . . . . . . . . . . . . 16 (𝑥 = (𝐻𝑣) → ((𝑆𝑥) ∈ (𝐾𝐶𝐵) ↔ (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵)))
6665ralrn 7108 . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . 13 (𝜑 → ∀𝑥 ∈ ran 𝐻(𝑆𝑥) ∈ (𝐾𝐶𝐵))
6968r19.21bi 3249 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ran 𝐻) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
7057, 69jaodan 959 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑂𝑥 ∈ ran 𝐻)) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
7156, 70sylan2b 594 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑂 ∪ ran 𝐻)) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
7271ralrimiva 3144 . . . . . . . . 9 (𝜑 → ∀𝑥 ∈ (𝑂 ∪ ran 𝐻)(𝑆𝑥) ∈ (𝐾𝐶𝐵))
73 mclsax.5 . . . . . . . . . . . 12 (𝜑𝑆 ∈ ran 𝐿)
7412, 5msubf 35517 . . . . . . . . . . . 12 (𝑆 ∈ ran 𝐿𝑆:𝐸𝐸)
7573, 74syl 17 . . . . . . . . . . 11 (𝜑𝑆:𝐸𝐸)
7675ffund 6741 . . . . . . . . . 10 (𝜑 → Fun 𝑆)
774, 5, 28elmpst 35521 . . . . . . . . . . . . . . 15 (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑀𝐷𝑀 = 𝑀) ∧ (𝑂𝐸𝑂 ∈ Fin) ∧ 𝑃𝐸))
7835, 77sylib 218 . . . . . . . . . . . . . 14 (𝜑 → ((𝑀𝐷𝑀 = 𝑀) ∧ (𝑂𝐸𝑂 ∈ Fin) ∧ 𝑃𝐸))
7978simp2d 1142 . . . . . . . . . . . . 13 (𝜑 → (𝑂𝐸𝑂 ∈ Fin))
8079simpld 494 . . . . . . . . . . . 12 (𝜑𝑂𝐸)
8175fdmd 6747 . . . . . . . . . . . 12 (𝜑 → dom 𝑆 = 𝐸)
8280, 81sseqtrrd 4037 . . . . . . . . . . 11 (𝜑𝑂 ⊆ dom 𝑆)
8362frnd 6745 . . . . . . . . . . . 12 (𝜑 → ran 𝐻𝐸)
8483, 81sseqtrrd 4037 . . . . . . . . . . 11 (𝜑 → ran 𝐻 ⊆ dom 𝑆)
8582, 84unssd 4202 . . . . . . . . . 10 (𝜑 → (𝑂 ∪ ran 𝐻) ⊆ dom 𝑆)
86 funimass4 6973 . . . . . . . . . 10 ((Fun 𝑆 ∧ (𝑂 ∪ ran 𝐻) ⊆ dom 𝑆) → ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ ∀𝑥 ∈ (𝑂 ∪ ran 𝐻)(𝑆𝑥) ∈ (𝐾𝐶𝐵)))
8776, 85, 86syl2anc 584 . . . . . . . . 9 (𝜑 → ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ ∀𝑥 ∈ (𝑂 ∪ ran 𝐻)(𝑆𝑥) ∈ (𝐾𝐶𝐵)))
8872, 87mpbird 257 . . . . . . . 8 (𝜑 → (𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵))
89 mclsax.8 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑀𝑦𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦))))) → 𝑎𝐾𝑏)
90893exp2 1353 . . . . . . . . . . . . 13 (𝜑 → (𝑥𝑀𝑦 → (𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) → (𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦))) → 𝑎𝐾𝑏))))
9190imp4b 421 . . . . . . . . . . . 12 ((𝜑𝑥𝑀𝑦) → ((𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦)))) → 𝑎𝐾𝑏))
9291ralrimivv 3198 . . . . . . . . . . 11 ((𝜑𝑥𝑀𝑦) → ∀𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥)))∀𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦)))𝑎𝐾𝑏)
93 dfss3 3984 . . . . . . . . . . . 12 (((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾 ↔ ∀𝑧 ∈ ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦))))𝑧𝐾)
94 eleq1 2827 . . . . . . . . . . . . . 14 (𝑧 = ⟨𝑎, 𝑏⟩ → (𝑧𝐾 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝐾))
95 df-br 5149 . . . . . . . . . . . . . 14 (𝑎𝐾𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝐾)
9694, 95bitr4di 289 . . . . . . . . . . . . 13 (𝑧 = ⟨𝑎, 𝑏⟩ → (𝑧𝐾𝑎𝐾𝑏))
9796ralxp 5855 . . . . . . . . . . . 12 (∀𝑧 ∈ ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦))))𝑧𝐾 ↔ ∀𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥)))∀𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦)))𝑎𝐾𝑏)
9893, 97bitri 275 . . . . . . . . . . 11 (((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾 ↔ ∀𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥)))∀𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦)))𝑎𝐾𝑏)
9992, 98sylibr 234 . . . . . . . . . 10 ((𝜑𝑥𝑀𝑦) → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾)
10099ex 412 . . . . . . . . 9 (𝜑 → (𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾))
101100alrimivv 1926 . . . . . . . 8 (𝜑 → ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾))
10288, 101jca 511 . . . . . . 7 (𝜑 → ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾)))
103 imaeq1 6075 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (𝑠 “ (𝑂 ∪ ran 𝐻)) = (𝑆 “ (𝑂 ∪ ran 𝐻)))
104103sseq1d 4027 . . . . . . . . . . 11 (𝑠 = 𝑆 → ((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ (𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵)))
105 fveq1 6906 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑆 → (𝑠‘(𝐻𝑥)) = (𝑆‘(𝐻𝑥)))
106105fveq2d 6911 . . . . . . . . . . . . . . 15 (𝑠 = 𝑆 → (𝑊‘(𝑠‘(𝐻𝑥))) = (𝑊‘(𝑆‘(𝐻𝑥))))
107 fveq1 6906 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑆 → (𝑠‘(𝐻𝑦)) = (𝑆‘(𝐻𝑦)))
108107fveq2d 6911 . . . . . . . . . . . . . . 15 (𝑠 = 𝑆 → (𝑊‘(𝑠‘(𝐻𝑦))) = (𝑊‘(𝑆‘(𝐻𝑦))))
109106, 108xpeq12d 5720 . . . . . . . . . . . . . 14 (𝑠 = 𝑆 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) = ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))))
110109sseq1d 4027 . . . . . . . . . . . . 13 (𝑠 = 𝑆 → (((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾 ↔ ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾))
111110imbi2d 340 . . . . . . . . . . . 12 (𝑠 = 𝑆 → ((𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾) ↔ (𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾)))
1121112albidv 1921 . . . . . . . . . . 11 (𝑠 = 𝑆 → (∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾) ↔ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾)))
113104, 112anbi12d 632 . . . . . . . . . 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))))
114 fveq1 6906 . . . . . . . . . . 11 (𝑠 = 𝑆 → (𝑠𝑃) = (𝑆𝑃))
115114eleq1d 2824 . . . . . . . . . 10 (𝑠 = 𝑆 → ((𝑠𝑃) ∈ 𝑐 ↔ (𝑆𝑃) ∈ 𝑐))
116113, 115imbi12d 344 . . . . . . . . 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)))
117116rspcv 3618 . . . . . . . 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)))
11873, 117syl 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)))
119102, 118mpid 44 . . . . . 6 (𝜑 → (∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑃) ∈ 𝑐) → (𝑆𝑃) ∈ 𝑐))
12033, 119embantd 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))
12127, 55, 1203syld 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))
122121alrimiv 1925 . . 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))
123 fvex 6920 . . . 4 (𝑆𝑃) ∈ V
124123elintab 4963 . . 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))
125122, 124sylibr 234 . 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)))})
126125, 14eleqtrrd 2842 1 (𝜑 → (𝑆𝑃) ∈ (𝐾𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086  wal 1535   = wceq 1537  wcel 2106  {cab 2712  wral 3059  Vcvv 3478  cun 3961  wss 3963  cop 4637  cotp 4639   cint 4951   class class class wbr 5148   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  Fincfn 8984  mVRcmvar 35446  mAxcmax 35450  mExcmex 35452  mDVcmdv 35453  mVarscmvrs 35454  mSubstcmsub 35456  mVHcmvh 35457  mPreStcmpst 35458  mStatcmsta 35460  mFScmfs 35461  mClscmcls 35462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-word 14550  df-concat 14606  df-s1 14631  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-0g 17488  df-gsum 17489  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-frmd 18875  df-mrex 35471  df-mex 35472  df-mrsub 35475  df-msub 35476  df-mvh 35477  df-mpst 35478  df-msr 35479  df-msta 35480  df-mfs 35481  df-mcls 35482
This theorem is referenced by:  mclsppslem  35568
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