| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | mclsval.d | . . 3
⊢ 𝐷 = (mDV‘𝑇) | 
| 2 |  | mclsval.e | . . 3
⊢ 𝐸 = (mEx‘𝑇) | 
| 3 |  | mclsval.c | . . 3
⊢ 𝐶 = (mCls‘𝑇) | 
| 4 |  | mclsval.1 | . . 3
⊢ (𝜑 → 𝑇 ∈ mFS) | 
| 5 |  | mclsval.2 | . . 3
⊢ (𝜑 → 𝐾 ⊆ 𝐷) | 
| 6 |  | mclsval.3 | . . 3
⊢ (𝜑 → 𝐵 ⊆ 𝐸) | 
| 7 |  | mclsax.h | . . 3
⊢ 𝐻 = (mVH‘𝑇) | 
| 8 |  | mclsax.a | . . 3
⊢ 𝐴 = (mAx‘𝑇) | 
| 9 |  | mclsax.l | . . 3
⊢ 𝐿 = (mSubst‘𝑇) | 
| 10 |  | mclsax.w | . . 3
⊢ 𝑊 = (mVars‘𝑇) | 
| 11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | mclsval 35569 | . 2
⊢ (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)))}) | 
| 12 |  | mclsind.4 | . . . . . . 7
⊢ (𝜑 → 𝐵 ⊆ 𝑄) | 
| 13 | 6, 12 | ssind 4240 | . . . . . 6
⊢ (𝜑 → 𝐵 ⊆ (𝐸 ∩ 𝑄)) | 
| 14 |  | mclsax.v | . . . . . . . . . . 11
⊢ 𝑉 = (mVR‘𝑇) | 
| 15 | 14, 2, 7 | mvhf 35564 | . . . . . . . . . 10
⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸) | 
| 16 | 4, 15 | syl 17 | . . . . . . . . 9
⊢ (𝜑 → 𝐻:𝑉⟶𝐸) | 
| 17 | 16 | ffnd 6736 | . . . . . . . 8
⊢ (𝜑 → 𝐻 Fn 𝑉) | 
| 18 | 16 | ffvelcdmda 7103 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ 𝐸) | 
| 19 |  | mclsind.5 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ 𝑄) | 
| 20 | 18, 19 | elind 4199 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ (𝐸 ∩ 𝑄)) | 
| 21 | 20 | ralrimiva 3145 | . . . . . . . 8
⊢ (𝜑 → ∀𝑣 ∈ 𝑉 (𝐻‘𝑣) ∈ (𝐸 ∩ 𝑄)) | 
| 22 |  | ffnfv 7138 | . . . . . . . 8
⊢ (𝐻:𝑉⟶(𝐸 ∩ 𝑄) ↔ (𝐻 Fn 𝑉 ∧ ∀𝑣 ∈ 𝑉 (𝐻‘𝑣) ∈ (𝐸 ∩ 𝑄))) | 
| 23 | 17, 21, 22 | sylanbrc 583 | . . . . . . 7
⊢ (𝜑 → 𝐻:𝑉⟶(𝐸 ∩ 𝑄)) | 
| 24 | 23 | frnd 6743 | . . . . . 6
⊢ (𝜑 → ran 𝐻 ⊆ (𝐸 ∩ 𝑄)) | 
| 25 | 13, 24 | unssd 4191 | . . . . 5
⊢ (𝜑 → (𝐵 ∪ ran 𝐻) ⊆ (𝐸 ∩ 𝑄)) | 
| 26 |  | id 22 | . . . . . . . . . . . 12
⊢ ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄)) | 
| 27 |  | inss2 4237 | . . . . . . . . . . . 12
⊢ (𝐸 ∩ 𝑄) ⊆ 𝑄 | 
| 28 | 26, 27 | sstrdi 3995 | . . . . . . . . . . 11
⊢ ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) | 
| 29 | 4 | adantr 480 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑇 ∈ mFS) | 
| 30 |  | eqid 2736 | . . . . . . . . . . . . . . . . . . . . 21
⊢
(mREx‘𝑇) =
(mREx‘𝑇) | 
| 31 | 14, 30, 9, 2 | msubff 35536 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑇 ∈ mFS → 𝐿:((mREx‘𝑇) ↑pm 𝑉)⟶(𝐸 ↑m 𝐸)) | 
| 32 |  | frn 6742 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝐿:((mREx‘𝑇) ↑pm 𝑉)⟶(𝐸 ↑m 𝐸) → ran 𝐿 ⊆ (𝐸 ↑m 𝐸)) | 
| 33 | 29, 31, 32 | 3syl 18 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → ran 𝐿 ⊆ (𝐸 ↑m 𝐸)) | 
| 34 |  | simpr2 1195 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑠 ∈ ran 𝐿) | 
| 35 | 33, 34 | sseldd 3983 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑠 ∈ (𝐸 ↑m 𝐸)) | 
| 36 |  | elmapi 8890 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ (𝐸 ↑m 𝐸) → 𝑠:𝐸⟶𝐸) | 
| 37 | 35, 36 | syl 17 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑠:𝐸⟶𝐸) | 
| 38 |  | eqid 2736 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
(mStat‘𝑇) =
(mStat‘𝑇) | 
| 39 | 8, 38 | maxsta 35560 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑇 ∈ mFS → 𝐴 ⊆ (mStat‘𝑇)) | 
| 40 | 29, 39 | syl 17 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝐴 ⊆ (mStat‘𝑇)) | 
| 41 |  | eqid 2736 | . . . . . . . . . . . . . . . . . . . . 21
⊢
(mPreSt‘𝑇) =
(mPreSt‘𝑇) | 
| 42 | 41, 38 | mstapst 35553 | . . . . . . . . . . . . . . . . . . . 20
⊢
(mStat‘𝑇)
⊆ (mPreSt‘𝑇) | 
| 43 | 40, 42 | sstrdi 3995 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝐴 ⊆ (mPreSt‘𝑇)) | 
| 44 |  | simpr1 1194 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴) | 
| 45 | 43, 44 | sseldd 3983 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 〈𝑚, 𝑜, 𝑝〉 ∈ (mPreSt‘𝑇)) | 
| 46 | 1, 2, 41 | elmpst 35542 | . . . . . . . . . . . . . . . . . . 19
⊢
(〈𝑚, 𝑜, 𝑝〉 ∈ (mPreSt‘𝑇) ↔ ((𝑚 ⊆ 𝐷 ∧ ◡𝑚 = 𝑚) ∧ (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin) ∧ 𝑝 ∈ 𝐸)) | 
| 47 | 46 | simp3bi 1147 | . . . . . . . . . . . . . . . . . 18
⊢
(〈𝑚, 𝑜, 𝑝〉 ∈ (mPreSt‘𝑇) → 𝑝 ∈ 𝐸) | 
| 48 | 45, 47 | syl 17 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑝 ∈ 𝐸) | 
| 49 | 37, 48 | ffvelcdmd 7104 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → (𝑠‘𝑝) ∈ 𝐸) | 
| 50 | 49 | 3adant3 1132 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸) | 
| 51 |  | mclsind.6 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑄) | 
| 52 | 50, 51 | elind 4199 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄)) | 
| 53 | 52 | 3exp 1119 | . . . . . . . . . . . . 13
⊢ (𝜑 → ((〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) → (∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄)))) | 
| 54 | 53 | 3expd 1353 | . . . . . . . . . . . 12
⊢ (𝜑 → (〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → (𝑠 ∈ ran 𝐿 → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄 → (∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄)))))) | 
| 55 | 54 | imp31 417 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴) ∧ 𝑠 ∈ ran 𝐿) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄 → (∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄)))) | 
| 56 | 28, 55 | syl5 34 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴) ∧ 𝑠 ∈ ran 𝐿) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) → (∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄)))) | 
| 57 | 56 | impd 410 | . . . . . . . . 9
⊢ (((𝜑 ∧ 〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴) ∧ 𝑠 ∈ ran 𝐿) → (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄))) | 
| 58 | 57 | ralrimiva 3145 | . . . . . . . 8
⊢ ((𝜑 ∧ 〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴) → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄))) | 
| 59 | 58 | ex 412 | . . . . . . 7
⊢ (𝜑 → (〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄)))) | 
| 60 | 59 | alrimiv 1926 | . . . . . 6
⊢ (𝜑 → ∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄)))) | 
| 61 | 60 | alrimivv 1927 | . . . . 5
⊢ (𝜑 → ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄)))) | 
| 62 | 2 | fvexi 6919 | . . . . . . 7
⊢ 𝐸 ∈ V | 
| 63 | 62 | inex1 5316 | . . . . . 6
⊢ (𝐸 ∩ 𝑄) ∈ V | 
| 64 |  | sseq2 4009 | . . . . . . 7
⊢ (𝑐 = (𝐸 ∩ 𝑄) → ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ↔ (𝐵 ∪ ran 𝐻) ⊆ (𝐸 ∩ 𝑄))) | 
| 65 |  | sseq2 4009 | . . . . . . . . . . . . 13
⊢ (𝑐 = (𝐸 ∩ 𝑄) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ↔ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄))) | 
| 66 | 65 | anbi1d 631 | . . . . . . . . . . . 12
⊢ (𝑐 = (𝐸 ∩ 𝑄) → (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) ↔ ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)))) | 
| 67 |  | eleq2 2829 | . . . . . . . . . . . 12
⊢ (𝑐 = (𝐸 ∩ 𝑄) → ((𝑠‘𝑝) ∈ 𝑐 ↔ (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄))) | 
| 68 | 66, 67 | imbi12d 344 | . . . . . . . . . . 11
⊢ (𝑐 = (𝐸 ∩ 𝑄) → ((((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐) ↔ (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄)))) | 
| 69 | 68 | ralbidv 3177 | . . . . . . . . . 10
⊢ (𝑐 = (𝐸 ∩ 𝑄) → (∀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)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄)))) | 
| 70 | 69 | imbi2d 340 | . . . . . . . . 9
⊢ (𝑐 = (𝐸 ∩ 𝑄) → ((〈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)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄))))) | 
| 71 | 70 | albidv 1919 | . . . . . . . 8
⊢ (𝑐 = (𝐸 ∩ 𝑄) → (∀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)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄))))) | 
| 72 | 71 | 2albidv 1922 | . . . . . . 7
⊢ (𝑐 = (𝐸 ∩ 𝑄) → (∀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)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄))))) | 
| 73 | 64, 72 | anbi12d 632 | . . . . . 6
⊢ (𝑐 = (𝐸 ∩ 𝑄) → (((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) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄)))))) | 
| 74 | 63, 73 | elab 3678 | . . . . 5
⊢ ((𝐸 ∩ 𝑄) ∈ {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) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄))))) | 
| 75 | 25, 61, 74 | sylanbrc 583 | . . . 4
⊢ (𝜑 → (𝐸 ∩ 𝑄) ∈ {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)))}) | 
| 76 |  | intss1 4962 | . . . 4
⊢ ((𝐸 ∩ 𝑄) ∈ {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)))} ⊆ (𝐸 ∩ 𝑄)) | 
| 77 | 75, 76 | syl 17 | . . 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)))} ⊆ (𝐸 ∩ 𝑄)) | 
| 78 | 77, 27 | sstrdi 3995 | . 2
⊢ (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)))} ⊆ 𝑄) | 
| 79 | 11, 78 | eqsstrd 4017 | 1
⊢ (𝜑 → (𝐾𝐶𝐵) ⊆ 𝑄) |