Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mclsax Structured version   Visualization version   GIF version

Theorem mclsax 34154
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 2717 . . . . . . . 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 4924 . . . . . . . 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 234 . . . . . . 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 34148 . . . . . . . 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 3975 . . . . . . 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 245 . . . . . 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 3951 . . . . . . . . . . . . . . 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 3166 . . . . . . . . . . 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 1919 . . . . . . . . 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 1921 . . . . . . . 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 2736 . . . . . . . 8 (mPreSt‘𝑇) = (mPreSt‘𝑇)
29 eqid 2736 . . . . . . . 8 (mStat‘𝑇) = (mStat‘𝑇)
3028, 29mstapst 34132 . . . . . . 7 (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
3111, 29maxsta 34139 . . . . . . . . 9 (𝑇 ∈ mFS → 𝐴 ⊆ (mStat‘𝑇))
327, 31syl 17 . . . . . . . 8 (𝜑𝐴 ⊆ (mStat‘𝑇))
33 mclsax.4 . . . . . . . 8 (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴)
3432, 33sseldd 3945 . . . . . . 7 (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mStat‘𝑇))
3530, 34sselid 3942 . . . . . 6 (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇))
3628mpstrcl 34126 . . . . . 6 (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) → (𝑀 ∈ V ∧ 𝑂 ∈ V ∧ 𝑃 ∈ V))
37 simp1 1136 . . . . . . . . . 10 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → 𝑚 = 𝑀)
38 simp2 1137 . . . . . . . . . 10 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → 𝑜 = 𝑂)
39 simp3 1138 . . . . . . . . . 10 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → 𝑝 = 𝑃)
4037, 38, 39oteq123d 4845 . . . . . . . . 9 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → ⟨𝑚, 𝑜, 𝑝⟩ = ⟨𝑀, 𝑂, 𝑃⟩)
4140eleq1d 2822 . . . . . . . 8 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ↔ ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴))
4238uneq1d 4122 . . . . . . . . . . . . 13 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (𝑜 ∪ ran 𝐻) = (𝑂 ∪ ran 𝐻))
4342imaeq2d 6013 . . . . . . . . . . . 12 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (𝑠 “ (𝑜 ∪ ran 𝐻)) = (𝑠 “ (𝑂 ∪ ran 𝐻)))
4443sseq1d 3975 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ (𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵)))
4537breqd 5116 . . . . . . . . . . . . 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 1926 . . . . . . . . . . 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 631 . . . . . . . . . 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 6846 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑜 = 𝑂𝑝 = 𝑃) → (𝑠𝑝) = (𝑠𝑃))
5049eleq1d 2822 . . . . . . . . . 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 3174 . . . . . . . 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 3563 . . . . . 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 4108 . . . . . . . . . . 11 (𝑥 ∈ (𝑂 ∪ ran 𝐻) ↔ (𝑥𝑂𝑥 ∈ ran 𝐻))
57 mclsax.6 . . . . . . . . . . . 12 ((𝜑𝑥𝑂) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
58 mclsax.7 . . . . . . . . . . . . . . 15 ((𝜑𝑣𝑉) → (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))
5958ralrimiva 3143 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑣𝑉 (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))
60 mclsax.v . . . . . . . . . . . . . . . . 17 𝑉 = (mVR‘𝑇)
6160, 5, 10mvhf 34143 . . . . . . . . . . . . . . . 16 (𝑇 ∈ mFS → 𝐻:𝑉𝐸)
627, 61syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐻:𝑉𝐸)
63 ffn 6668 . . . . . . . . . . . . . . 15 (𝐻:𝑉𝐸𝐻 Fn 𝑉)
64 fveq2 6842 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝐻𝑣) → (𝑆𝑥) = (𝑆‘(𝐻𝑣)))
6564eleq1d 2822 . . . . . . . . . . . . . . . 16 (𝑥 = (𝐻𝑣) → ((𝑆𝑥) ∈ (𝐾𝐶𝐵) ↔ (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵)))
6665ralrn 7037 . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . 13 (𝜑 → ∀𝑥 ∈ ran 𝐻(𝑆𝑥) ∈ (𝐾𝐶𝐵))
6968r19.21bi 3234 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ran 𝐻) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
7057, 69jaodan 956 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑂𝑥 ∈ ran 𝐻)) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
7156, 70sylan2b 594 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑂 ∪ ran 𝐻)) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
7271ralrimiva 3143 . . . . . . . . 9 (𝜑 → ∀𝑥 ∈ (𝑂 ∪ ran 𝐻)(𝑆𝑥) ∈ (𝐾𝐶𝐵))
73 mclsax.5 . . . . . . . . . . . 12 (𝜑𝑆 ∈ ran 𝐿)
7412, 5msubf 34117 . . . . . . . . . . . 12 (𝑆 ∈ ran 𝐿𝑆:𝐸𝐸)
7573, 74syl 17 . . . . . . . . . . 11 (𝜑𝑆:𝐸𝐸)
7675ffund 6672 . . . . . . . . . 10 (𝜑 → Fun 𝑆)
774, 5, 28elmpst 34121 . . . . . . . . . . . . . . 15 (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑀𝐷𝑀 = 𝑀) ∧ (𝑂𝐸𝑂 ∈ Fin) ∧ 𝑃𝐸))
7835, 77sylib 217 . . . . . . . . . . . . . 14 (𝜑 → ((𝑀𝐷𝑀 = 𝑀) ∧ (𝑂𝐸𝑂 ∈ Fin) ∧ 𝑃𝐸))
7978simp2d 1143 . . . . . . . . . . . . 13 (𝜑 → (𝑂𝐸𝑂 ∈ Fin))
8079simpld 495 . . . . . . . . . . . 12 (𝜑𝑂𝐸)
8175fdmd 6679 . . . . . . . . . . . 12 (𝜑 → dom 𝑆 = 𝐸)
8280, 81sseqtrrd 3985 . . . . . . . . . . 11 (𝜑𝑂 ⊆ dom 𝑆)
8362frnd 6676 . . . . . . . . . . . 12 (𝜑 → ran 𝐻𝐸)
8483, 81sseqtrrd 3985 . . . . . . . . . . 11 (𝜑 → ran 𝐻 ⊆ dom 𝑆)
8582, 84unssd 4146 . . . . . . . . . 10 (𝜑 → (𝑂 ∪ ran 𝐻) ⊆ dom 𝑆)
86 funimass4 6907 . . . . . . . . . 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 256 . . . . . . . 8 (𝜑 → (𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵))
89 mclsax.8 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑀𝑦𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦))))) → 𝑎𝐾𝑏)
90893exp2 1354 . . . . . . . . . . . . 13 (𝜑 → (𝑥𝑀𝑦 → (𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) → (𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦))) → 𝑎𝐾𝑏))))
9190imp4b 422 . . . . . . . . . . . 12 ((𝜑𝑥𝑀𝑦) → ((𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦)))) → 𝑎𝐾𝑏))
9291ralrimivv 3195 . . . . . . . . . . 11 ((𝜑𝑥𝑀𝑦) → ∀𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥)))∀𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦)))𝑎𝐾𝑏)
93 dfss3 3932 . . . . . . . . . . . 12 (((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾 ↔ ∀𝑧 ∈ ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦))))𝑧𝐾)
94 eleq1 2825 . . . . . . . . . . . . . 14 (𝑧 = ⟨𝑎, 𝑏⟩ → (𝑧𝐾 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝐾))
95 df-br 5106 . . . . . . . . . . . . . 14 (𝑎𝐾𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝐾)
9694, 95bitr4di 288 . . . . . . . . . . . . 13 (𝑧 = ⟨𝑎, 𝑏⟩ → (𝑧𝐾𝑎𝐾𝑏))
9796ralxp 5797 . . . . . . . . . . . 12 (∀𝑧 ∈ ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦))))𝑧𝐾 ↔ ∀𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥)))∀𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦)))𝑎𝐾𝑏)
9893, 97bitri 274 . . . . . . . . . . 11 (((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾 ↔ ∀𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥)))∀𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦)))𝑎𝐾𝑏)
9992, 98sylibr 233 . . . . . . . . . 10 ((𝜑𝑥𝑀𝑦) → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾)
10099ex 413 . . . . . . . . 9 (𝜑 → (𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾))
101100alrimivv 1931 . . . . . . . 8 (𝜑 → ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾))
10288, 101jca 512 . . . . . . 7 (𝜑 → ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾)))
103 imaeq1 6008 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (𝑠 “ (𝑂 ∪ ran 𝐻)) = (𝑆 “ (𝑂 ∪ ran 𝐻)))
104103sseq1d 3975 . . . . . . . . . . 11 (𝑠 = 𝑆 → ((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ (𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵)))
105 fveq1 6841 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑆 → (𝑠‘(𝐻𝑥)) = (𝑆‘(𝐻𝑥)))
106105fveq2d 6846 . . . . . . . . . . . . . . 15 (𝑠 = 𝑆 → (𝑊‘(𝑠‘(𝐻𝑥))) = (𝑊‘(𝑆‘(𝐻𝑥))))
107 fveq1 6841 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑆 → (𝑠‘(𝐻𝑦)) = (𝑆‘(𝐻𝑦)))
108107fveq2d 6846 . . . . . . . . . . . . . . 15 (𝑠 = 𝑆 → (𝑊‘(𝑠‘(𝐻𝑦))) = (𝑊‘(𝑆‘(𝐻𝑦))))
109106, 108xpeq12d 5664 . . . . . . . . . . . . . 14 (𝑠 = 𝑆 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) = ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))))
110109sseq1d 3975 . . . . . . . . . . . . 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 1926 . . . . . . . . . . 11 (𝑠 = 𝑆 → (∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾) ↔ ∀𝑥𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻𝑥))) × (𝑊‘(𝑆‘(𝐻𝑦)))) ⊆ 𝐾)))
113104, 112anbi12d 631 . . . . . . . . . 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 6841 . . . . . . . . . . 11 (𝑠 = 𝑆 → (𝑠𝑃) = (𝑆𝑃))
115114eleq1d 2822 . . . . . . . . . 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 3577 . . . . . . . 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 1930 . . 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 6855 . . . 4 (𝑆𝑃) ∈ V
124123elintab 4919 . . 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 233 . 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 2841 1 (𝜑 → (𝑆𝑃) ∈ (𝐾𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845  w3a 1087  wal 1539   = wceq 1541  wcel 2106  {cab 2713  wral 3064  Vcvv 3445  cun 3908  wss 3910  cop 4592  cotp 4594   cint 4907   class class class wbr 5105   × cxp 5631  ccnv 5632  dom cdm 5633  ran crn 5634  cima 5636  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7356  Fincfn 8882  mVRcmvar 34046  mAxcmax 34050  mExcmex 34052  mDVcmdv 34053  mVarscmvrs 34054  mSubstcmsub 34056  mVHcmvh 34057  mPreStcmpst 34058  mStatcmsta 34060  mFScmfs 34061  mClscmcls 34062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671  ax-cnex 11106  ax-resscn 11107  ax-1cn 11108  ax-icn 11109  ax-addcl 11110  ax-addrcl 11111  ax-mulcl 11112  ax-mulrcl 11113  ax-mulcom 11114  ax-addass 11115  ax-mulass 11116  ax-distr 11117  ax-i2m1 11118  ax-1ne0 11119  ax-1rid 11120  ax-rnegex 11121  ax-rrecex 11122  ax-cnre 11123  ax-pre-lttri 11124  ax-pre-lttrn 11125  ax-pre-ltadd 11126  ax-pre-mulgt0 11127
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-ot 4595  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7312  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7802  df-1st 7920  df-2nd 7921  df-frecs 8211  df-wrecs 8242  df-recs 8316  df-rdg 8355  df-1o 8411  df-er 8647  df-map 8766  df-pm 8767  df-en 8883  df-dom 8884  df-sdom 8885  df-fin 8886  df-card 9874  df-pnf 11190  df-mnf 11191  df-xr 11192  df-ltxr 11193  df-le 11194  df-sub 11386  df-neg 11387  df-nn 12153  df-2 12215  df-n0 12413  df-z 12499  df-uz 12763  df-fz 13424  df-fzo 13567  df-seq 13906  df-hash 14230  df-word 14402  df-concat 14458  df-s1 14483  df-struct 17018  df-sets 17035  df-slot 17053  df-ndx 17065  df-base 17083  df-ress 17112  df-plusg 17145  df-0g 17322  df-gsum 17323  df-mgm 18496  df-sgrp 18545  df-mnd 18556  df-submnd 18601  df-frmd 18658  df-mrex 34071  df-mex 34072  df-mrsub 34075  df-msub 34076  df-mvh 34077  df-mpst 34078  df-msr 34079  df-msta 34080  df-mfs 34081  df-mcls 34082
This theorem is referenced by:  mclsppslem  34168
  Copyright terms: Public domain W3C validator