Proof of Theorem mclsssvlem
Step | Hyp | Ref
| Expression |
1 | | mclsval.3 |
. . . 4
⊢ (𝜑 → 𝐵 ⊆ 𝐸) |
2 | | mclsval.1 |
. . . . . 6
⊢ (𝜑 → 𝑇 ∈ mFS) |
3 | | eqid 2610 |
. . . . . . 7
⊢
(mVR‘𝑇) =
(mVR‘𝑇) |
4 | | mclsval.e |
. . . . . . 7
⊢ 𝐸 = (mEx‘𝑇) |
5 | | mclsval.h |
. . . . . . 7
⊢ 𝐻 = (mVH‘𝑇) |
6 | 3, 4, 5 | mvhf 30709 |
. . . . . 6
⊢ (𝑇 ∈ mFS → 𝐻:(mVR‘𝑇)⟶𝐸) |
7 | 2, 6 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝐻:(mVR‘𝑇)⟶𝐸) |
8 | | frn 5966 |
. . . . 5
⊢ (𝐻:(mVR‘𝑇)⟶𝐸 → ran 𝐻 ⊆ 𝐸) |
9 | 7, 8 | syl 17 |
. . . 4
⊢ (𝜑 → ran 𝐻 ⊆ 𝐸) |
10 | 1, 9 | unssd 3751 |
. . 3
⊢ (𝜑 → (𝐵 ∪ ran 𝐻) ⊆ 𝐸) |
11 | | mclsval.s |
. . . . . . . . . 10
⊢ 𝑆 = (mSubst‘𝑇) |
12 | 11, 4 | msubf 30683 |
. . . . . . . . 9
⊢ (𝑠 ∈ ran 𝑆 → 𝑠:𝐸⟶𝐸) |
13 | | mclsval.a |
. . . . . . . . . . . . . 14
⊢ 𝐴 = (mAx‘𝑇) |
14 | | eqid 2610 |
. . . . . . . . . . . . . 14
⊢
(mStat‘𝑇) =
(mStat‘𝑇) |
15 | 13, 14 | maxsta 30705 |
. . . . . . . . . . . . 13
⊢ (𝑇 ∈ mFS → 𝐴 ⊆ (mStat‘𝑇)) |
16 | 2, 15 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ⊆ (mStat‘𝑇)) |
17 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢
(mPreSt‘𝑇) =
(mPreSt‘𝑇) |
18 | 17, 14 | mstapst 30698 |
. . . . . . . . . . . 12
⊢
(mStat‘𝑇)
⊆ (mPreSt‘𝑇) |
19 | 16, 18 | syl6ss 3580 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ⊆ (mPreSt‘𝑇)) |
20 | 19 | sselda 3568 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴) → 〈𝑚, 𝑜, 𝑝〉 ∈ (mPreSt‘𝑇)) |
21 | | mclsval.d |
. . . . . . . . . . . 12
⊢ 𝐷 = (mDV‘𝑇) |
22 | 21, 4, 17 | elmpst 30687 |
. . . . . . . . . . 11
⊢
(〈𝑚, 𝑜, 𝑝〉 ∈ (mPreSt‘𝑇) ↔ ((𝑚 ⊆ 𝐷 ∧ ◡𝑚 = 𝑚) ∧ (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin) ∧ 𝑝 ∈ 𝐸)) |
23 | 22 | simp3bi 1071 |
. . . . . . . . . 10
⊢
(〈𝑚, 𝑜, 𝑝〉 ∈ (mPreSt‘𝑇) → 𝑝 ∈ 𝐸) |
24 | 20, 23 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴) → 𝑝 ∈ 𝐸) |
25 | | ffvelrn 6265 |
. . . . . . . . 9
⊢ ((𝑠:𝐸⟶𝐸 ∧ 𝑝 ∈ 𝐸) → (𝑠‘𝑝) ∈ 𝐸) |
26 | 12, 24, 25 | syl2anr 494 |
. . . . . . . 8
⊢ (((𝜑 ∧ 〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴) ∧ 𝑠 ∈ ran 𝑆) → (𝑠‘𝑝) ∈ 𝐸) |
27 | 26 | a1d 25 |
. . . . . . 7
⊢ (((𝜑 ∧ 〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴) ∧ 𝑠 ∈ ran 𝑆) → (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸)) |
28 | 27 | ralrimiva 2949 |
. . . . . 6
⊢ ((𝜑 ∧ 〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴) → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸)) |
29 | 28 | ex 449 |
. . . . 5
⊢ (𝜑 → (〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸))) |
30 | 29 | alrimiv 1842 |
. . . 4
⊢ (𝜑 → ∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸))) |
31 | 30 | alrimivv 1843 |
. . 3
⊢ (𝜑 → ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸))) |
32 | | fvex 6113 |
. . . . 5
⊢
(mEx‘𝑇) ∈
V |
33 | 4, 32 | eqeltri 2684 |
. . . 4
⊢ 𝐸 ∈ V |
34 | | sseq2 3590 |
. . . . 5
⊢ (𝑐 = 𝐸 → ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ↔ (𝐵 ∪ ran 𝐻) ⊆ 𝐸)) |
35 | | sseq2 3590 |
. . . . . . . . . . 11
⊢ (𝑐 = 𝐸 → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ↔ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸)) |
36 | 35 | anbi1d 737 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐸 → (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) ↔ ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)))) |
37 | | eleq2 2677 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐸 → ((𝑠‘𝑝) ∈ 𝑐 ↔ (𝑠‘𝑝) ∈ 𝐸)) |
38 | 36, 37 | imbi12d 333 |
. . . . . . . . 9
⊢ (𝑐 = 𝐸 → ((((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐) ↔ (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸))) |
39 | 38 | ralbidv 2969 |
. . . . . . . 8
⊢ (𝑐 = 𝐸 → (∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐) ↔ ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸))) |
40 | 39 | imbi2d 329 |
. . . . . . 7
⊢ (𝑐 = 𝐸 → ((〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) ↔ (〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸)))) |
41 | 40 | albidv 1836 |
. . . . . 6
⊢ (𝑐 = 𝐸 → (∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) ↔ ∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸)))) |
42 | 41 | 2albidv 1838 |
. . . . 5
⊢ (𝑐 = 𝐸 → (∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) ↔ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸)))) |
43 | 34, 42 | anbi12d 743 |
. . . 4
⊢ (𝑐 = 𝐸 → (((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) ↔ ((𝐵 ∪ ran 𝐻) ⊆ 𝐸 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸))))) |
44 | 33, 43 | elab 3319 |
. . 3
⊢ (𝐸 ∈ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ↔ ((𝐵 ∪ ran 𝐻) ⊆ 𝐸 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸)))) |
45 | 10, 31, 44 | sylanbrc 695 |
. 2
⊢ (𝜑 → 𝐸 ∈ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))}) |
46 | | intss1 4427 |
. 2
⊢ (𝐸 ∈ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} → ∩
{𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ 𝐸) |
47 | 45, 46 | syl 17 |
1
⊢ (𝜑 → ∩ {𝑐
∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ 𝐸) |