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Theorem mclsax 31803
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 2759 . . . . . . . 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 4627 . . . . . . . 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 31797 . . . . . . . 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 3781 . . . . . . 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 3759 . . . . . . . . . . . . . . 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 598 . . . . . . . . . . . . 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 3112 . . . . . . . . . . 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 1997 . . . . . . . . 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 1999 . . . . . . . 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 467 . . . . . 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 2771 . . . . . . . 8 (mPreSt‘𝑇) = (mPreSt‘𝑇)
29 eqid 2771 . . . . . . . 8 (mStat‘𝑇) = (mStat‘𝑇)
3028, 29mstapst 31781 . . . . . . 7 (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
3111, 29maxsta 31788 . . . . . . . . 9 (𝑇 ∈ mFS → 𝐴 ⊆ (mStat‘𝑇))
327, 31syl 17 . . . . . . . 8 (𝜑𝐴 ⊆ (mStat‘𝑇))
33 mclsax.4 . . . . . . . 8 (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴)
3432, 33sseldd 3753 . . . . . . 7 (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mStat‘𝑇))
3530, 34sseldi 3750 . . . . . 6 (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇))
3628mpstrcl 31775 . . . . . 6 (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) → (𝑀 ∈ V ∧ 𝑂 ∈ V ∧ 𝑃 ∈ V))
37 simp1 1130 . . . . . . . . . 10 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → 𝑚 = 𝑀)
38 simp2 1131 . . . . . . . . . 10 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → 𝑜 = 𝑂)
39 simp3 1132 . . . . . . . . . 10 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → 𝑝 = 𝑃)
4037, 38, 39oteq123d 4555 . . . . . . . . 9 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → ⟨𝑚, 𝑜, 𝑝⟩ = ⟨𝑀, 𝑂, 𝑃⟩)
4140eleq1d 2835 . . . . . . . 8 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ↔ ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴))
4238uneq1d 3917 . . . . . . . . . . . . 13 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (𝑜 ∪ ran 𝐻) = (𝑂 ∪ ran 𝐻))
4342imaeq2d 5606 . . . . . . . . . . . 12 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (𝑠 “ (𝑜 ∪ ran 𝐻)) = (𝑠 “ (𝑂 ∪ ran 𝐻)))
4443sseq1d 3781 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ (𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵)))
4537breqd 4798 . . . . . . . . . . . . 13 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (𝑥𝑚𝑦𝑥𝑀𝑦))
4645imbi1d 330 . . . . . . . . . . . 12 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → ((𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾) ↔ (𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)))
47462albidv 2003 . . . . . . . . . . 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 616 . . . . . . . . . 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 6337 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (𝑠𝑝) = (𝑠𝑃))
5049eleq1d 2835 . . . . . . . . . 10 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → ((𝑠𝑝) ∈ 𝑐 ↔ (𝑠𝑃) ∈ 𝑐))
5148, 50imbi12d 333 . . . . . . . . 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 3135 . . . . . . . 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 333 . . . . . . 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 3449 . . . . . 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 3904 . . . . . . . . . . 11 (𝑥 ∈ (𝑂 ∪ ran 𝐻) ↔ (𝑥𝑂𝑥 ∈ ran 𝐻))
57 mclsax.6 . . . . . . . . . . . 12 ((𝜑𝑥𝑂) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
58 mclsax.7 . . . . . . . . . . . . . . 15 ((𝜑𝑣𝑉) → (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))
5958ralrimiva 3115 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑣𝑉 (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))
60 mclsax.v . . . . . . . . . . . . . . . . 17 𝑉 = (mVR‘𝑇)
6160, 5, 10mvhf 31792 . . . . . . . . . . . . . . . 16 (𝑇 ∈ mFS → 𝐻:𝑉𝐸)
627, 61syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐻:𝑉𝐸)
63 ffn 6184 . . . . . . . . . . . . . . 15 (𝐻:𝑉𝐸𝐻 Fn 𝑉)
64 fveq2 6333 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝐻𝑣) → (𝑆𝑥) = (𝑆‘(𝐻𝑣)))
6564eleq1d 2835 . . . . . . . . . . . . . . . 16 (𝑥 = (𝐻𝑣) → ((𝑆𝑥) ∈ (𝐾𝐶𝐵) ↔ (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵)))
6665ralrn 6507 . . . . . . . . . . . . . . 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 3081 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ran 𝐻) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
7057, 69jaodan 942 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑂𝑥 ∈ ran 𝐻)) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
7156, 70sylan2b 581 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑂 ∪ ran 𝐻)) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
7271ralrimiva 3115 . . . . . . . . 9 (𝜑 → ∀𝑥 ∈ (𝑂 ∪ ran 𝐻)(𝑆𝑥) ∈ (𝐾𝐶𝐵))
73 mclsax.5 . . . . . . . . . . . 12 (𝜑𝑆 ∈ ran 𝐿)
7412, 5msubf 31766 . . . . . . . . . . . 12 (𝑆 ∈ ran 𝐿𝑆:𝐸𝐸)
7573, 74syl 17 . . . . . . . . . . 11 (𝜑𝑆:𝐸𝐸)
7675ffund 6188 . . . . . . . . . 10 (𝜑 → Fun 𝑆)
774, 5, 28elmpst 31770 . . . . . . . . . . . . . . 15 (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑀𝐷𝑀 = 𝑀) ∧ (𝑂𝐸𝑂 ∈ Fin) ∧ 𝑃𝐸))
7835, 77sylib 208 . . . . . . . . . . . . . 14 (𝜑 → ((𝑀𝐷𝑀 = 𝑀) ∧ (𝑂𝐸𝑂 ∈ Fin) ∧ 𝑃𝐸))
7978simp2d 1137 . . . . . . . . . . . . 13 (𝜑 → (𝑂𝐸𝑂 ∈ Fin))
8079simpld 482 . . . . . . . . . . . 12 (𝜑𝑂𝐸)
8175fdmd 6193 . . . . . . . . . . . 12 (𝜑 → dom 𝑆 = 𝐸)
8280, 81sseqtr4d 3791 . . . . . . . . . . 11 (𝜑𝑂 ⊆ dom 𝑆)
8362frnd 6191 . . . . . . . . . . . 12 (𝜑 → ran 𝐻𝐸)
8483, 81sseqtr4d 3791 . . . . . . . . . . 11 (𝜑 → ran 𝐻 ⊆ dom 𝑆)
8582, 84unssd 3940 . . . . . . . . . 10 (𝜑 → (𝑂 ∪ ran 𝐻) ⊆ dom 𝑆)
86 funimass4 6391 . . . . . . . . . 10 ((Fun 𝑆 ∧ (𝑂 ∪ ran 𝐻) ⊆ dom 𝑆) → ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ ∀𝑥 ∈ (𝑂 ∪ ran 𝐻)(𝑆𝑥) ∈ (𝐾𝐶𝐵)))
8776, 85, 86syl2anc 573 . . . . . . . . 9 (𝜑 → ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ ∀𝑥 ∈ (𝑂 ∪ ran 𝐻)(𝑆𝑥) ∈ (𝐾𝐶𝐵)))
8872, 87mpbird 247 . . . . . . . 8 (𝜑 → (𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵))
89 mclsax.8 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑀𝑦𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦))))) → 𝑎𝐾𝑏)
90893exp2 1447 . . . . . . . . . . . . 13 (𝜑 → (𝑥𝑀𝑦 → (𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) → (𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦))) → 𝑎𝐾𝑏))))
9190imp4b 408 . . . . . . . . . . . 12 ((𝜑𝑥𝑀𝑦) → ((𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦)))) → 𝑎𝐾𝑏))
9291ralrimivv 3119 . . . . . . . . . . 11 ((𝜑𝑥𝑀𝑦) → ∀𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥)))∀𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦)))𝑎𝐾𝑏)
93 dfss3 3741 . . . . . . . . . . . 12 (((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾 ↔ ∀𝑧 ∈ ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦))))𝑧𝐾)
94 eleq1 2838 . . . . . . . . . . . . . 14 (𝑧 = ⟨𝑎, 𝑏⟩ → (𝑧𝐾 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝐾))
95 df-br 4788 . . . . . . . . . . . . . 14 (𝑎𝐾𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝐾)
9694, 95syl6bbr 278 . . . . . . . . . . . . 13 (𝑧 = ⟨𝑎, 𝑏⟩ → (𝑧𝐾𝑎𝐾𝑏))
9796ralxp 5401 . . . . . . . . . . . 12 (∀𝑧 ∈ ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦))))𝑧𝐾 ↔ ∀𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥)))∀𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦)))𝑎𝐾𝑏)
9893, 97bitri 264 . . . . . . . . . . 11 (((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾 ↔ ∀𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥)))∀𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦)))𝑎𝐾𝑏)
9992, 98sylibr 224 . . . . . . . . . 10 ((𝜑𝑥𝑀𝑦) → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾)
10099ex 397 . . . . . . . . 9 (𝜑 → (𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾))
101100alrimivv 2008 . . . . . . . 8 (𝜑 → ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾))
10288, 101jca 501 . . . . . . 7 (𝜑 → ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾)))
103 imaeq1 5601 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (𝑠 “ (𝑂 ∪ ran 𝐻)) = (𝑆 “ (𝑂 ∪ ran 𝐻)))
104103sseq1d 3781 . . . . . . . . . . 11 (𝑠 = 𝑆 → ((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ (𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵)))
105 fveq1 6332 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑆 → (𝑠‘(𝐻𝑥)) = (𝑆‘(𝐻𝑥)))
106105fveq2d 6337 . . . . . . . . . . . . . . 15 (𝑠 = 𝑆 → (𝑊‘(𝑠‘(𝐻𝑥))) = (𝑊‘(𝑆‘(𝐻𝑥))))
107 fveq1 6332 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑆 → (𝑠‘(𝐻𝑦)) = (𝑆‘(𝐻𝑦)))
108107fveq2d 6337 . . . . . . . . . . . . . . 15 (𝑠 = 𝑆 → (𝑊‘(𝑠‘(𝐻𝑦))) = (𝑊‘(𝑆‘(𝐻𝑦))))
109106, 108xpeq12d 5280 . . . . . . . . . . . . . 14 (𝑠 = 𝑆 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) = ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))))
110109sseq1d 3781 . . . . . . . . . . . . 13 (𝑠 = 𝑆 → (((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾 ↔ ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾))
111110imbi2d 329 . . . . . . . . . . . 12 (𝑠 = 𝑆 → ((𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾) ↔ (𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾)))
1121112albidv 2003 . . . . . . . . . . 11 (𝑠 = 𝑆 → (∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾) ↔ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾)))
113104, 112anbi12d 616 . . . . . . . . . 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 6332 . . . . . . . . . . 11 (𝑠 = 𝑆 → (𝑠𝑃) = (𝑆𝑃))
115114eleq1d 2835 . . . . . . . . . 10 (𝑠 = 𝑆 → ((𝑠𝑃) ∈ 𝑐 ↔ (𝑆𝑃) ∈ 𝑐))
116113, 115imbi12d 333 . . . . . . . . 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 3456 . . . . . . . 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 2007 . . 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 6344 . . . 4 (𝑆𝑃) ∈ V
124123elintab 4623 . . 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 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)))})
126125, 14eleqtrrd 2853 1 (𝜑 → (𝑆𝑃) ∈ (𝐾𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wo 836  w3a 1071  wal 1629   = wceq 1631  wcel 2145  {cab 2757  wral 3061  Vcvv 3351  cun 3721  wss 3723  cop 4323  cotp 4325   cint 4612   class class class wbr 4787   × cxp 5248  ccnv 5249  dom cdm 5250  ran crn 5251  cima 5253  Fun wfun 6024   Fn wfn 6025  wf 6026  cfv 6030  (class class class)co 6795  Fincfn 8112  mVRcmvar 31695  mAxcmax 31699  mExcmex 31701  mDVcmdv 31702  mVarscmvrs 31703  mSubstcmsub 31705  mVHcmvh 31706  mPreStcmpst 31707  mStatcmsta 31709  mFScmfs 31710  mClscmcls 31711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099  ax-cnex 10197  ax-resscn 10198  ax-1cn 10199  ax-icn 10200  ax-addcl 10201  ax-addrcl 10202  ax-mulcl 10203  ax-mulrcl 10204  ax-mulcom 10205  ax-addass 10206  ax-mulass 10207  ax-distr 10208  ax-i2m1 10209  ax-1ne0 10210  ax-1rid 10211  ax-rnegex 10212  ax-rrecex 10213  ax-cnre 10214  ax-pre-lttri 10215  ax-pre-lttrn 10216  ax-pre-ltadd 10217  ax-pre-mulgt0 10218
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-ot 4326  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6756  df-ov 6798  df-oprab 6799  df-mpt2 6800  df-om 7216  df-1st 7318  df-2nd 7319  df-wrecs 7562  df-recs 7624  df-rdg 7662  df-1o 7716  df-oadd 7720  df-er 7899  df-map 8014  df-pm 8015  df-en 8113  df-dom 8114  df-sdom 8115  df-fin 8116  df-card 8968  df-pnf 10281  df-mnf 10282  df-xr 10283  df-ltxr 10284  df-le 10285  df-sub 10473  df-neg 10474  df-nn 11226  df-2 11284  df-n0 11499  df-z 11584  df-uz 11893  df-fz 12533  df-fzo 12673  df-seq 13008  df-hash 13321  df-word 13494  df-concat 13496  df-s1 13497  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-plusg 16161  df-0g 16309  df-gsum 16310  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-submnd 17543  df-frmd 17593  df-mrex 31720  df-mex 31721  df-mrsub 31724  df-msub 31725  df-mvh 31726  df-mpst 31727  df-msr 31728  df-msta 31729  df-mfs 31730  df-mcls 31731
This theorem is referenced by:  mclsppslem  31817
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