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Theorem mclsax 35574
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 2718 . . . . . . . 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 4963 . . . . . . . 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 35568 . . . . . . . 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 4015 . . . . . . 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 3990 . . . . . . . . . . . . . . 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 3169 . . . . . . . . . . 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 1916 . . . . . . . . 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 1918 . . . . . . . 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 2737 . . . . . . . 8 (mPreSt‘𝑇) = (mPreSt‘𝑇)
29 eqid 2737 . . . . . . . 8 (mStat‘𝑇) = (mStat‘𝑇)
3028, 29mstapst 35552 . . . . . . 7 (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
3111, 29maxsta 35559 . . . . . . . . 9 (𝑇 ∈ mFS → 𝐴 ⊆ (mStat‘𝑇))
327, 31syl 17 . . . . . . . 8 (𝜑𝐴 ⊆ (mStat‘𝑇))
33 mclsax.4 . . . . . . . 8 (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴)
3432, 33sseldd 3984 . . . . . . 7 (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mStat‘𝑇))
3530, 34sselid 3981 . . . . . 6 (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇))
3628mpstrcl 35546 . . . . . 6 (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) → (𝑀 ∈ V ∧ 𝑂 ∈ V ∧ 𝑃 ∈ V))
37 simp1 1137 . . . . . . . . . 10 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → 𝑚 = 𝑀)
38 simp2 1138 . . . . . . . . . 10 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → 𝑜 = 𝑂)
39 simp3 1139 . . . . . . . . . 10 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → 𝑝 = 𝑃)
4037, 38, 39oteq123d 4888 . . . . . . . . 9 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → ⟨𝑚, 𝑜, 𝑝⟩ = ⟨𝑀, 𝑂, 𝑃⟩)
4140eleq1d 2826 . . . . . . . 8 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ↔ ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴))
4238uneq1d 4167 . . . . . . . . . . . . 13 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (𝑜 ∪ ran 𝐻) = (𝑂 ∪ ran 𝐻))
4342imaeq2d 6078 . . . . . . . . . . . 12 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (𝑠 “ (𝑜 ∪ ran 𝐻)) = (𝑠 “ (𝑂 ∪ ran 𝐻)))
4443sseq1d 4015 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ (𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵)))
4537breqd 5154 . . . . . . . . . . . . 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 1923 . . . . . . . . . . 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 6910 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (𝑠𝑝) = (𝑠𝑃))
5049eleq1d 2826 . . . . . . . . . 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 3178 . . . . . . . 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 4153 . . . . . . . . . . 11 (𝑥 ∈ (𝑂 ∪ ran 𝐻) ↔ (𝑥𝑂𝑥 ∈ ran 𝐻))
57 mclsax.6 . . . . . . . . . . . 12 ((𝜑𝑥𝑂) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
58 mclsax.7 . . . . . . . . . . . . . . 15 ((𝜑𝑣𝑉) → (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))
5958ralrimiva 3146 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑣𝑉 (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))
60 mclsax.v . . . . . . . . . . . . . . . . 17 𝑉 = (mVR‘𝑇)
6160, 5, 10mvhf 35563 . . . . . . . . . . . . . . . 16 (𝑇 ∈ mFS → 𝐻:𝑉𝐸)
627, 61syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐻:𝑉𝐸)
63 ffn 6736 . . . . . . . . . . . . . . 15 (𝐻:𝑉𝐸𝐻 Fn 𝑉)
64 fveq2 6906 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝐻𝑣) → (𝑆𝑥) = (𝑆‘(𝐻𝑣)))
6564eleq1d 2826 . . . . . . . . . . . . . . . 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 3251 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ran 𝐻) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
7057, 69jaodan 960 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑂𝑥 ∈ ran 𝐻)) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
7156, 70sylan2b 594 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑂 ∪ ran 𝐻)) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
7271ralrimiva 3146 . . . . . . . . 9 (𝜑 → ∀𝑥 ∈ (𝑂 ∪ ran 𝐻)(𝑆𝑥) ∈ (𝐾𝐶𝐵))
73 mclsax.5 . . . . . . . . . . . 12 (𝜑𝑆 ∈ ran 𝐿)
7412, 5msubf 35537 . . . . . . . . . . . 12 (𝑆 ∈ ran 𝐿𝑆:𝐸𝐸)
7573, 74syl 17 . . . . . . . . . . 11 (𝜑𝑆:𝐸𝐸)
7675ffund 6740 . . . . . . . . . 10 (𝜑 → Fun 𝑆)
774, 5, 28elmpst 35541 . . . . . . . . . . . . . . 15 (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑀𝐷𝑀 = 𝑀) ∧ (𝑂𝐸𝑂 ∈ Fin) ∧ 𝑃𝐸))
7835, 77sylib 218 . . . . . . . . . . . . . 14 (𝜑 → ((𝑀𝐷𝑀 = 𝑀) ∧ (𝑂𝐸𝑂 ∈ Fin) ∧ 𝑃𝐸))
7978simp2d 1144 . . . . . . . . . . . . 13 (𝜑 → (𝑂𝐸𝑂 ∈ Fin))
8079simpld 494 . . . . . . . . . . . 12 (𝜑𝑂𝐸)
8175fdmd 6746 . . . . . . . . . . . 12 (𝜑 → dom 𝑆 = 𝐸)
8280, 81sseqtrrd 4021 . . . . . . . . . . 11 (𝜑𝑂 ⊆ dom 𝑆)
8362frnd 6744 . . . . . . . . . . . 12 (𝜑 → ran 𝐻𝐸)
8483, 81sseqtrrd 4021 . . . . . . . . . . 11 (𝜑 → ran 𝐻 ⊆ dom 𝑆)
8582, 84unssd 4192 . . . . . . . . . 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 1355 . . . . . . . . . . . . 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 3200 . . . . . . . . . . 11 ((𝜑𝑥𝑀𝑦) → ∀𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥)))∀𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦)))𝑎𝐾𝑏)
93 dfss3 3972 . . . . . . . . . . . 12 (((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾 ↔ ∀𝑧 ∈ ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦))))𝑧𝐾)
94 eleq1 2829 . . . . . . . . . . . . . 14 (𝑧 = ⟨𝑎, 𝑏⟩ → (𝑧𝐾 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝐾))
95 df-br 5144 . . . . . . . . . . . . . 14 (𝑎𝐾𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝐾)
9694, 95bitr4di 289 . . . . . . . . . . . . 13 (𝑧 = ⟨𝑎, 𝑏⟩ → (𝑧𝐾𝑎𝐾𝑏))
9796ralxp 5852 . . . . . . . . . . . 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 1928 . . . . . . . 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 6073 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (𝑠 “ (𝑂 ∪ ran 𝐻)) = (𝑆 “ (𝑂 ∪ ran 𝐻)))
104103sseq1d 4015 . . . . . . . . . . 11 (𝑠 = 𝑆 → ((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ (𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵)))
105 fveq1 6905 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑆 → (𝑠‘(𝐻𝑥)) = (𝑆‘(𝐻𝑥)))
106105fveq2d 6910 . . . . . . . . . . . . . . 15 (𝑠 = 𝑆 → (𝑊‘(𝑠‘(𝐻𝑥))) = (𝑊‘(𝑆‘(𝐻𝑥))))
107 fveq1 6905 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑆 → (𝑠‘(𝐻𝑦)) = (𝑆‘(𝐻𝑦)))
108107fveq2d 6910 . . . . . . . . . . . . . . 15 (𝑠 = 𝑆 → (𝑊‘(𝑠‘(𝐻𝑦))) = (𝑊‘(𝑆‘(𝐻𝑦))))
109106, 108xpeq12d 5716 . . . . . . . . . . . . . 14 (𝑠 = 𝑆 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) = ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))))
110109sseq1d 4015 . . . . . . . . . . . . 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 1923 . . . . . . . . . . 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 6905 . . . . . . . . . . 11 (𝑠 = 𝑆 → (𝑠𝑃) = (𝑆𝑃))
115114eleq1d 2826 . . . . . . . . . 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 1927 . . 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 6919 . . . 4 (𝑆𝑃) ∈ V
124123elintab 4958 . . 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 2844 1 (𝜑 → (𝑆𝑃) ∈ (𝐾𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  w3a 1087  wal 1538   = wceq 1540  wcel 2108  {cab 2714  wral 3061  Vcvv 3480  cun 3949  wss 3951  cop 4632  cotp 4634   cint 4946   class class class wbr 5143   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  cima 5688  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  Fincfn 8985  mVRcmvar 35466  mAxcmax 35470  mExcmex 35472  mDVcmdv 35473  mVarscmvrs 35474  mSubstcmsub 35476  mVHcmvh 35477  mPreStcmpst 35478  mStatcmsta 35480  mFScmfs 35481  mClscmcls 35482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-word 14553  df-concat 14609  df-s1 14634  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-gsum 17487  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-frmd 18862  df-mrex 35491  df-mex 35492  df-mrsub 35495  df-msub 35496  df-mvh 35497  df-mpst 35498  df-msr 35499  df-msta 35500  df-mfs 35501  df-mcls 35502
This theorem is referenced by:  mclsppslem  35588
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