Theorem mclsax 35767

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	⊢ (𝜑 → 𝑇 ∈ mFS)
mclsval.2	⊢ (𝜑 → 𝐾 ⊆ 𝐷)
mclsval.3	⊢ (𝜑 → 𝐵 ⊆ 𝐸)
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	⊢ (𝜑 → 𝑆 ∈ ran 𝐿)
mclsax.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
mclsax.7	⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))
mclsax.8	⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏)

Assertion

Ref	Expression
mclsax	⊢ (𝜑 → (𝑆‘𝑃) ∈ (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.

Step	Hyp	Ref	Expression
1		abid 2719	. . . . . . . 8 ⊢ (𝑐 ∈ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ↔ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))))
2		intss1 4906	. . . . . . . 8 ⊢ (𝑐 ∈ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} → ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ 𝑐)
3	1, 2	sylbir 235	. . . . . . 7 ⊢ (((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) → ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 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 ⊢ (𝜑 → 𝑇 ∈ mFS)
8		mclsval.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ⊆ 𝐷)
9		mclsval.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ 𝐸)
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)
14	4, 5, 6, 7, 8, 9, 10, 11, 12, 13	mclsval 35761	. . . . . . . 8 ⊢ (𝜑 → (𝐾𝐶𝐵) = ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))})
15	14	sseq1d 3954	. . . . . . 7 ⊢ (𝜑 → ((𝐾𝐶𝐵) ⊆ 𝑐 ↔ ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ 𝑐))
16	3, 15	imbitrrid 246	. . . . . 6 ⊢ (𝜑 → (((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) → (𝐾𝐶𝐵) ⊆ 𝑐))
17		sstr2 3929	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) → ((𝐾𝐶𝐵) ⊆ 𝑐 → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐))
18	17	com12 32	. . . . . . . . . . . . . 14 ⊢ ((𝐾𝐶𝐵) ⊆ 𝑐 → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐))
19	18	anim1d 612	. . . . . . . . . . . . 13 ⊢ ((𝐾𝐶𝐵) ⊆ 𝑐 → (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾))))
20	19	imim1d 82	. . . . . . . . . . . 12 ⊢ ((𝐾𝐶𝐵) ⊆ 𝑐 → ((((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐) → (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))
21	20	ralimdv 3152	. . . . . . . . . . 11 ⊢ ((𝐾𝐶𝐵) ⊆ 𝑐 → (∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐) → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))
22	21	imim2d 57	. . . . . . . . . 10 ⊢ ((𝐾𝐶𝐵) ⊆ 𝑐 → ((⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) → (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))))
23	22	alimdv 1918	. . . . . . . . 9 ⊢ ((𝐾𝐶𝐵) ⊆ 𝑐 → (∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) → ∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))))
24	23	2alimdv 1920	. . . . . . . 8 ⊢ ((𝐾𝐶𝐵) ⊆ 𝑐 → (∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) → ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))))
25	24	com12 32	. . . . . . 7 ⊢ (∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) → ((𝐾𝐶𝐵) ⊆ 𝑐 → ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))))
26	25	adantl 481	. . . . . 6 ⊢ (((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) → ((𝐾𝐶𝐵) ⊆ 𝑐 → ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))))
27	16, 26	sylcom 30	. . . . 5 ⊢ (𝜑 → (((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) → ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))))
28		eqid 2737	. . . . . . . 8 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
29		eqid 2737	. . . . . . . 8 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
30	28, 29	mstapst 35745	. . . . . . 7 ⊢ (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
31	11, 29	maxsta 35752	. . . . . . . . 9 ⊢ (𝑇 ∈ mFS → 𝐴 ⊆ (mStat‘𝑇))
32	7, 31	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ (mStat‘𝑇))
33		mclsax.4	. . . . . . . 8 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴)
34	32, 33	sseldd 3923	. . . . . . 7 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mStat‘𝑇))
35	30, 34	sselid 3920	. . . . . 6 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇))
36	28	mpstrcl 35739	. . . . . 6 ⊢ (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) → (𝑀 ∈ V ∧ 𝑂 ∈ V ∧ 𝑃 ∈ V))
37		simp1 1137	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → 𝑚 = 𝑀)
38		simp2 1138	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → 𝑜 = 𝑂)
39		simp3 1139	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃)
40	37, 38, 39	oteq123d 4832	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → ⟨𝑚, 𝑜, 𝑝⟩ = ⟨𝑀, 𝑂, 𝑃⟩)
41	40	eleq1d 2822	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ↔ ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴))
42	38	uneq1d 4108	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (𝑜 ∪ ran 𝐻) = (𝑂 ∪ ran 𝐻))
43	42	imaeq2d 6019	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (𝑠 “ (𝑜 ∪ ran 𝐻)) = (𝑠 “ (𝑂 ∪ ran 𝐻)))
44	43	sseq1d 3954	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ (𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵)))
45	37	breqd 5097	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (𝑥𝑚𝑦 ↔ 𝑥𝑀𝑦))
46	45	imbi1d 341	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → ((𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾) ↔ (𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)))
47	46	2albidv 1925	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾) ↔ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)))
48	44, 47	anbi12d 633	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) ↔ ((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾))))
49	39	fveq2d 6838	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (𝑠‘𝑝) = (𝑠‘𝑃))
50	49	eleq1d 2822	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → ((𝑠‘𝑝) ∈ 𝑐 ↔ (𝑠‘𝑃) ∈ 𝑐))
51	48, 50	imbi12d 344	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → ((((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐) ↔ (((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑃) ∈ 𝑐)))
52	51	ralbidv 3161	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐) ↔ ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑃) ∈ 𝑐)))
53	41, 52	imbi12d 344	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → ((⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) ↔ (⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑃) ∈ 𝑐))))
54	53	spc3gv 3547	. . . . . 6 ⊢ ((𝑀 ∈ V ∧ 𝑂 ∈ V ∧ 𝑃 ∈ V) → (∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) → (⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑃) ∈ 𝑐))))
55	35, 36, 54	3syl 18	. . . . 5 ⊢ (𝜑 → (∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) → (⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑃) ∈ 𝑐))))
56		elun 4094	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑂 ∪ ran 𝐻) ↔ (𝑥 ∈ 𝑂 ∨ 𝑥 ∈ ran 𝐻))
57		mclsax.6	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
58		mclsax.7	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))
59	58	ralrimiva 3130	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑣 ∈ 𝑉 (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))
60		mclsax.v	. . . . . . . . . . . . . . . . 17 ⊢ 𝑉 = (mVR‘𝑇)
61	60, 5, 10	mvhf 35756	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸)
62	7, 61	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻:𝑉⟶𝐸)
63		ffn 6662	. . . . . . . . . . . . . . 15 ⊢ (𝐻:𝑉⟶𝐸 → 𝐻 Fn 𝑉)
64		fveq2 6834	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝐻‘𝑣) → (𝑆‘𝑥) = (𝑆‘(𝐻‘𝑣)))
65	64	eleq1d 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝐻‘𝑣) → ((𝑆‘𝑥) ∈ (𝐾𝐶𝐵) ↔ (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵)))
66	65	ralrn 7034	. . . . . . . . . . . . . . 15 ⊢ (𝐻 Fn 𝑉 → (∀𝑥 ∈ ran 𝐻(𝑆‘𝑥) ∈ (𝐾𝐶𝐵) ↔ ∀𝑣 ∈ 𝑉 (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵)))
67	62, 63, 66	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐻(𝑆‘𝑥) ∈ (𝐾𝐶𝐵) ↔ ∀𝑣 ∈ 𝑉 (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵)))
68	59, 67	mpbird 257	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐻(𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
69	68	r19.21bi 3230	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐻) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
70	57, 69	jaodan 960	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑂 ∨ 𝑥 ∈ ran 𝐻)) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
71	56, 70	sylan2b 595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑂 ∪ ran 𝐻)) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
72	71	ralrimiva 3130	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ (𝑂 ∪ ran 𝐻)(𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
73		mclsax.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ ran 𝐿)
74	12, 5	msubf 35730	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ ran 𝐿 → 𝑆:𝐸⟶𝐸)
75	73, 74	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆:𝐸⟶𝐸)
76	75	ffund 6666	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝑆)
77	4, 5, 28	elmpst 35734	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑀 ⊆ 𝐷 ∧ ◡𝑀 = 𝑀) ∧ (𝑂 ⊆ 𝐸 ∧ 𝑂 ∈ Fin) ∧ 𝑃 ∈ 𝐸))
78	35, 77	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑀 ⊆ 𝐷 ∧ ◡𝑀 = 𝑀) ∧ (𝑂 ⊆ 𝐸 ∧ 𝑂 ∈ Fin) ∧ 𝑃 ∈ 𝐸))
79	78	simp2d 1144	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂 ⊆ 𝐸 ∧ 𝑂 ∈ Fin))
80	79	simpld 494	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑂 ⊆ 𝐸)
81	75	fdmd 6672	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝑆 = 𝐸)
82	80, 81	sseqtrrd 3960	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑂 ⊆ dom 𝑆)
83	62	frnd 6670	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐻 ⊆ 𝐸)
84	83, 81	sseqtrrd 3960	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝐻 ⊆ dom 𝑆)
85	82, 84	unssd 4133	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂 ∪ ran 𝐻) ⊆ dom 𝑆)
86		funimass4 6898	. . . . . . . . . 10 ⊢ ((Fun 𝑆 ∧ (𝑂 ∪ ran 𝐻) ⊆ dom 𝑆) → ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ ∀𝑥 ∈ (𝑂 ∪ ran 𝐻)(𝑆‘𝑥) ∈ (𝐾𝐶𝐵)))
87	76, 85, 86	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ ∀𝑥 ∈ (𝑂 ∪ ran 𝐻)(𝑆‘𝑥) ∈ (𝐾𝐶𝐵)))
88	72, 87	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → (𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵))
89		mclsax.8	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏)
90	89	3exp2 1356	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥𝑀𝑦 → (𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) → (𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))) → 𝑎𝐾𝑏))))
91	90	imp4b 421	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥𝑀𝑦) → ((𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦)))) → 𝑎𝐾𝑏))
92	91	ralrimivv 3179	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥𝑀𝑦) → ∀𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥)))∀𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦)))𝑎𝐾𝑏)
93		dfss3 3911	. . . . . . . . . . . 12 ⊢ (((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾 ↔ ∀𝑧 ∈ ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦))))𝑧 ∈ 𝐾)
94		eleq1 2825	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑎, 𝑏⟩ → (𝑧 ∈ 𝐾 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝐾))
95		df-br 5087	. . . . . . . . . . . . . 14 ⊢ (𝑎𝐾𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝐾)
96	94, 95	bitr4di 289	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑎, 𝑏⟩ → (𝑧 ∈ 𝐾 ↔ 𝑎𝐾𝑏))
97	96	ralxp 5790	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦))))𝑧 ∈ 𝐾 ↔ ∀𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥)))∀𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦)))𝑎𝐾𝑏)
98	93, 97	bitri 275	. . . . . . . . . . 11 ⊢ (((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾 ↔ ∀𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥)))∀𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦)))𝑎𝐾𝑏)
99	92, 98	sylibr 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥𝑀𝑦) → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾)
100	99	ex 412	. . . . . . . . 9 ⊢ (𝜑 → (𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾))
101	100	alrimivv 1930	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾))
102	88, 101	jca 511	. . . . . . 7 ⊢ (𝜑 → ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾)))
103		imaeq1 6014	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (𝑠 “ (𝑂 ∪ ran 𝐻)) = (𝑆 “ (𝑂 ∪ ran 𝐻)))
104	103	sseq1d 3954	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → ((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ (𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵)))
105		fveq1 6833	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑆 → (𝑠‘(𝐻‘𝑥)) = (𝑆‘(𝐻‘𝑥)))
106	105	fveq2d 6838	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑆 → (𝑊‘(𝑠‘(𝐻‘𝑥))) = (𝑊‘(𝑆‘(𝐻‘𝑥))))
107		fveq1 6833	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑆 → (𝑠‘(𝐻‘𝑦)) = (𝑆‘(𝐻‘𝑦)))
108	107	fveq2d 6838	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑆 → (𝑊‘(𝑠‘(𝐻‘𝑦))) = (𝑊‘(𝑆‘(𝐻‘𝑦))))
109	106, 108	xpeq12d 5655	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑆 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) = ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))))
110	109	sseq1d 3954	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑆 → (((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾 ↔ ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾))
111	110	imbi2d 340	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → ((𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾) ↔ (𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾)))
112	111	2albidv 1925	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾) ↔ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾)))
113	104, 112	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) ↔ ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾))))
114		fveq1 6833	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (𝑠‘𝑃) = (𝑆‘𝑃))
115	114	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → ((𝑠‘𝑃) ∈ 𝑐 ↔ (𝑆‘𝑃) ∈ 𝑐))
116	113, 115	imbi12d 344	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → ((((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑃) ∈ 𝑐) ↔ (((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑆‘𝑃) ∈ 𝑐)))
117	116	rspcv 3561	. . . . . . . 8 ⊢ (𝑆 ∈ ran 𝐿 → (∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑃) ∈ 𝑐) → (((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑆‘𝑃) ∈ 𝑐)))
118	73, 117	syl 17	. . . . . . 7 ⊢ (𝜑 → (∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑃) ∈ 𝑐) → (((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑆‘𝑃) ∈ 𝑐)))
119	102, 118	mpid 44	. . . . . 6 ⊢ (𝜑 → (∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑃) ∈ 𝑐) → (𝑆‘𝑃) ∈ 𝑐))
120	33, 119	embantd 59	. . . . 5 ⊢ (𝜑 → ((⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑃) ∈ 𝑐)) → (𝑆‘𝑃) ∈ 𝑐))
121	27, 55, 120	3syld 60	. . . 4 ⊢ (𝜑 → (((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) → (𝑆‘𝑃) ∈ 𝑐))
122	121	alrimiv 1929	. . 3 ⊢ (𝜑 → ∀𝑐(((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) → (𝑆‘𝑃) ∈ 𝑐))
123		fvex 6847	. . . 4 ⊢ (𝑆‘𝑃) ∈ V
124	123	elintab 4902	. . 3 ⊢ ((𝑆‘𝑃) ∈ ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ↔ ∀𝑐(((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) → (𝑆‘𝑃) ∈ 𝑐))
125	122, 124	sylibr 234	. 2 ⊢ (𝜑 → (𝑆‘𝑃) ∈ ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))})
126	125, 14	eleqtrrd 2840	1 ⊢ (𝜑 → (𝑆‘𝑃) ∈ (𝐾𝐶𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087  ∀wal 1540   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  Vcvv 3430   ∪ cun 3888   ⊆ wss 3890  ⟨cop 4574  ⟨cotp 4576  ∩ cint 4890   class class class wbr 5086   × cxp 5622  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   “ cima 5627  Fun wfun 6486   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  Fincfn 8886  mVRcmvar 35659  mAxcmax 35663  mExcmex 35665  mDVcmdv 35666  mVarscmvrs 35667  mSubstcmsub 35669  mVHcmvh 35670  mPreStcmpst 35671  mStatcmsta 35673  mFScmfs 35674  mClscmcls 35675

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-map 8768  df-pm 8769  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-n0 12429  df-z 12516  df-uz 12780  df-fz 13453  df-fzo 13600  df-seq 13955  df-hash 14284  df-word 14467  df-concat 14524  df-s1 14550  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-0g 17395  df-gsum 17396  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-submnd 18743  df-frmd 18808  df-mrex 35684  df-mex 35685  df-mrsub 35688  df-msub 35689  df-mvh 35690  df-mpst 35691  df-msr 35692  df-msta 35693  df-mfs 35694  df-mcls 35695

This theorem is referenced by:  mclsppslem  35781