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Theorem mthmpps 34176
Description: Given a theorem, there is an explicitly definable witnessing provable pre-statement for the provability of the theorem. (However, this pre-statement requires infinitely many disjoint variable conditions, which is sometimes inconvenient.) (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mthmpps.r 𝑅 = (mStRed‘𝑇)
mthmpps.j 𝐽 = (mPPSt‘𝑇)
mthmpps.u 𝑈 = (mThm‘𝑇)
mthmpps.d 𝐷 = (mDV‘𝑇)
mthmpps.v 𝑉 = (mVars‘𝑇)
mthmpps.z 𝑍 = (𝑉 “ (𝐻 ∪ {𝐴}))
mthmpps.m 𝑀 = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍)))
Assertion
Ref Expression
mthmpps (𝑇 ∈ mFS → (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 ↔ (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))))

Proof of Theorem mthmpps
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 mthmpps.m . . . . . . . 8 𝑀 = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍)))
2 mthmpps.u . . . . . . . . . . . . . 14 𝑈 = (mThm‘𝑇)
3 eqid 2736 . . . . . . . . . . . . . 14 (mPreSt‘𝑇) = (mPreSt‘𝑇)
42, 3mthmsta 34172 . . . . . . . . . . . . 13 𝑈 ⊆ (mPreSt‘𝑇)
5 simpr 485 . . . . . . . . . . . . 13 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈)
64, 5sselid 3942 . . . . . . . . . . . 12 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇))
7 mthmpps.d . . . . . . . . . . . . 13 𝐷 = (mDV‘𝑇)
8 eqid 2736 . . . . . . . . . . . . 13 (mEx‘𝑇) = (mEx‘𝑇)
97, 8, 3elmpst 34130 . . . . . . . . . . . 12 (⟨𝐶, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ↔ ((𝐶𝐷𝐶 = 𝐶) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
106, 9sylib 217 . . . . . . . . . . 11 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ((𝐶𝐷𝐶 = 𝐶) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
1110simp1d 1142 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶𝐷𝐶 = 𝐶))
1211simpld 495 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐶𝐷)
13 difssd 4092 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐷 ∖ (𝑍 × 𝑍)) ⊆ 𝐷)
1412, 13unssd 4146 . . . . . . . 8 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ⊆ 𝐷)
151, 14eqsstrid 3992 . . . . . . 7 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝑀𝐷)
1611simprd 496 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐶 = 𝐶)
17 cnvdif 6096 . . . . . . . . . . 11 (𝐷 ∖ (𝑍 × 𝑍)) = (𝐷(𝑍 × 𝑍))
18 cnvdif 6096 . . . . . . . . . . . . . 14 (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
19 cnvxp 6109 . . . . . . . . . . . . . . 15 ((mVR‘𝑇) × (mVR‘𝑇)) = ((mVR‘𝑇) × (mVR‘𝑇))
20 cnvi 6094 . . . . . . . . . . . . . . 15 I = I
2119, 20difeq12i 4080 . . . . . . . . . . . . . 14 (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
2218, 21eqtri 2764 . . . . . . . . . . . . 13 (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
23 eqid 2736 . . . . . . . . . . . . . . 15 (mVR‘𝑇) = (mVR‘𝑇)
2423, 7mdvval 34098 . . . . . . . . . . . . . 14 𝐷 = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
2524cnveqi 5830 . . . . . . . . . . . . 13 𝐷 = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
2622, 25, 243eqtr4i 2774 . . . . . . . . . . . 12 𝐷 = 𝐷
27 cnvxp 6109 . . . . . . . . . . . 12 (𝑍 × 𝑍) = (𝑍 × 𝑍)
2826, 27difeq12i 4080 . . . . . . . . . . 11 (𝐷(𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍))
2917, 28eqtri 2764 . . . . . . . . . 10 (𝐷 ∖ (𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍))
3029a1i 11 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐷 ∖ (𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍)))
3116, 30uneq12d 4124 . . . . . . . 8 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶(𝐷 ∖ (𝑍 × 𝑍))) = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))))
321cnveqi 5830 . . . . . . . . 9 𝑀 = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍)))
33 cnvun 6095 . . . . . . . . 9 (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) = (𝐶(𝐷 ∖ (𝑍 × 𝑍)))
3432, 33eqtri 2764 . . . . . . . 8 𝑀 = (𝐶(𝐷 ∖ (𝑍 × 𝑍)))
3531, 34, 13eqtr4g 2801 . . . . . . 7 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝑀 = 𝑀)
3615, 35jca 512 . . . . . 6 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑀𝐷𝑀 = 𝑀))
3710simp2d 1143 . . . . . 6 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin))
3810simp3d 1144 . . . . . 6 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐴 ∈ (mEx‘𝑇))
397, 8, 3elmpst 34130 . . . . . 6 (⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑀𝐷𝑀 = 𝑀) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
4036, 37, 38, 39syl3anbrc 1343 . . . . 5 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇))
41 mthmpps.r . . . . . . . 8 𝑅 = (mStRed‘𝑇)
42 mthmpps.j . . . . . . . 8 𝐽 = (mPPSt‘𝑇)
4341, 42, 2elmthm 34170 . . . . . . 7 (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
445, 43sylib 217 . . . . . 6 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ∃𝑥𝐽 (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
45 eqid 2736 . . . . . . . 8 (mCls‘𝑇) = (mCls‘𝑇)
46 simpll 765 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑇 ∈ mFS)
4715adantr 481 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑀𝐷)
4837simpld 495 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐻 ⊆ (mEx‘𝑇))
4948adantr 481 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐻 ⊆ (mEx‘𝑇))
503, 42mppspst 34168 . . . . . . . . . . . . . . . . . . 19 𝐽 ⊆ (mPreSt‘𝑇)
51 simprl 769 . . . . . . . . . . . . . . . . . . 19 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥𝐽)
5250, 51sselid 3942 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 ∈ (mPreSt‘𝑇))
533mpst123 34134 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (mPreSt‘𝑇) → 𝑥 = ⟨(1st ‘(1st𝑥)), (2nd ‘(1st𝑥)), (2nd𝑥)⟩)
5452, 53syl 17 . . . . . . . . . . . . . . . . 17 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 = ⟨(1st ‘(1st𝑥)), (2nd ‘(1st𝑥)), (2nd𝑥)⟩)
5554fveq2d 6846 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅𝑥) = (𝑅‘⟨(1st ‘(1st𝑥)), (2nd ‘(1st𝑥)), (2nd𝑥)⟩))
56 simprr 771 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
5755, 56eqtr3d 2778 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅‘⟨(1st ‘(1st𝑥)), (2nd ‘(1st𝑥)), (2nd𝑥)⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
5854, 52eqeltrrd 2839 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1st ‘(1st𝑥)), (2nd ‘(1st𝑥)), (2nd𝑥)⟩ ∈ (mPreSt‘𝑇))
59 mthmpps.v . . . . . . . . . . . . . . . . 17 𝑉 = (mVars‘𝑇)
60 eqid 2736 . . . . . . . . . . . . . . . . 17 (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) = (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)}))
6159, 3, 41, 60msrval 34132 . . . . . . . . . . . . . . . 16 (⟨(1st ‘(1st𝑥)), (2nd ‘(1st𝑥)), (2nd𝑥)⟩ ∈ (mPreSt‘𝑇) → (𝑅‘⟨(1st ‘(1st𝑥)), (2nd ‘(1st𝑥)), (2nd𝑥)⟩) = ⟨((1st ‘(1st𝑥)) ∩ ( (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) × (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})))), (2nd ‘(1st𝑥)), (2nd𝑥)⟩)
6258, 61syl 17 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅‘⟨(1st ‘(1st𝑥)), (2nd ‘(1st𝑥)), (2nd𝑥)⟩) = ⟨((1st ‘(1st𝑥)) ∩ ( (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) × (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})))), (2nd ‘(1st𝑥)), (2nd𝑥)⟩)
63 mthmpps.z . . . . . . . . . . . . . . . . . 18 𝑍 = (𝑉 “ (𝐻 ∪ {𝐴}))
6459, 3, 41, 63msrval 34132 . . . . . . . . . . . . . . . . 17 (⟨𝐶, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) → (𝑅‘⟨𝐶, 𝐻, 𝐴⟩) = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
656, 64syl 17 . . . . . . . . . . . . . . . 16 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑅‘⟨𝐶, 𝐻, 𝐴⟩) = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
6665adantr 481 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅‘⟨𝐶, 𝐻, 𝐴⟩) = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
6757, 62, 663eqtr3d 2784 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨((1st ‘(1st𝑥)) ∩ ( (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) × (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})))), (2nd ‘(1st𝑥)), (2nd𝑥)⟩ = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
68 fvex 6855 . . . . . . . . . . . . . . . 16 (1st ‘(1st𝑥)) ∈ V
6968inex1 5274 . . . . . . . . . . . . . . 15 ((1st ‘(1st𝑥)) ∩ ( (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) × (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})))) ∈ V
70 fvex 6855 . . . . . . . . . . . . . . 15 (2nd ‘(1st𝑥)) ∈ V
71 fvex 6855 . . . . . . . . . . . . . . 15 (2nd𝑥) ∈ V
7269, 70, 71otth 5441 . . . . . . . . . . . . . 14 (⟨((1st ‘(1st𝑥)) ∩ ( (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) × (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})))), (2nd ‘(1st𝑥)), (2nd𝑥)⟩ = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ ↔ (((1st ‘(1st𝑥)) ∩ ( (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) × (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})))) = (𝐶 ∩ (𝑍 × 𝑍)) ∧ (2nd ‘(1st𝑥)) = 𝐻 ∧ (2nd𝑥) = 𝐴))
7367, 72sylib 217 . . . . . . . . . . . . 13 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (((1st ‘(1st𝑥)) ∩ ( (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) × (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})))) = (𝐶 ∩ (𝑍 × 𝑍)) ∧ (2nd ‘(1st𝑥)) = 𝐻 ∧ (2nd𝑥) = 𝐴))
7473simp1d 1142 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st𝑥)) ∩ ( (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) × (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})))) = (𝐶 ∩ (𝑍 × 𝑍)))
7573simp2d 1143 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (2nd ‘(1st𝑥)) = 𝐻)
7673simp3d 1144 . . . . . . . . . . . . . . . . . . 19 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (2nd𝑥) = 𝐴)
7776sneqd 4598 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → {(2nd𝑥)} = {𝐴})
7875, 77uneq12d 4124 . . . . . . . . . . . . . . . . 17 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)}) = (𝐻 ∪ {𝐴}))
7978imaeq2d 6013 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) = (𝑉 “ (𝐻 ∪ {𝐴})))
8079unieqd 4879 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) = (𝑉 “ (𝐻 ∪ {𝐴})))
8180, 63eqtr4di 2794 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) = 𝑍)
8281sqxpeqd 5665 . . . . . . . . . . . . 13 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ( (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) × (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)}))) = (𝑍 × 𝑍))
8382ineq2d 4172 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st𝑥)) ∩ ( (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) × (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})))) = ((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)))
8474, 83eqtr3d 2778 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝐶 ∩ (𝑍 × 𝑍)) = ((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)))
85 inss1 4188 . . . . . . . . . . 11 (𝐶 ∩ (𝑍 × 𝑍)) ⊆ 𝐶
8684, 85eqsstrrdi 3999 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)) ⊆ 𝐶)
87 eqidd 2737 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (1st ‘(1st𝑥)) = (1st ‘(1st𝑥)))
8887, 75, 76oteq123d 4845 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1st ‘(1st𝑥)), (2nd ‘(1st𝑥)), (2nd𝑥)⟩ = ⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩)
8954, 88eqtrd 2776 . . . . . . . . . . . . 13 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 = ⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩)
9089, 52eqeltrrd 2839 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇))
917, 8, 3elmpst 34130 . . . . . . . . . . . . . 14 (⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ↔ (((1st ‘(1st𝑥)) ⊆ 𝐷(1st ‘(1st𝑥)) = (1st ‘(1st𝑥))) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
9291simp1bi 1145 . . . . . . . . . . . . 13 (⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) → ((1st ‘(1st𝑥)) ⊆ 𝐷(1st ‘(1st𝑥)) = (1st ‘(1st𝑥))))
9392simpld 495 . . . . . . . . . . . 12 (⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) → (1st ‘(1st𝑥)) ⊆ 𝐷)
9490, 93syl 17 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (1st ‘(1st𝑥)) ⊆ 𝐷)
9594ssdifd 4100 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st𝑥)) ∖ (𝑍 × 𝑍)) ⊆ (𝐷 ∖ (𝑍 × 𝑍)))
96 unss12 4142 . . . . . . . . . 10 ((((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)) ⊆ 𝐶 ∧ ((1st ‘(1st𝑥)) ∖ (𝑍 × 𝑍)) ⊆ (𝐷 ∖ (𝑍 × 𝑍))) → (((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1st ‘(1st𝑥)) ∖ (𝑍 × 𝑍))) ⊆ (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))))
9786, 95, 96syl2anc 584 . . . . . . . . 9 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1st ‘(1st𝑥)) ∖ (𝑍 × 𝑍))) ⊆ (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))))
98 inundif 4438 . . . . . . . . . 10 (((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1st ‘(1st𝑥)) ∖ (𝑍 × 𝑍))) = (1st ‘(1st𝑥))
9998eqcomi 2745 . . . . . . . . 9 (1st ‘(1st𝑥)) = (((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1st ‘(1st𝑥)) ∖ (𝑍 × 𝑍)))
10097, 99, 13sstr4g 3989 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (1st ‘(1st𝑥)) ⊆ 𝑀)
101 ssidd 3967 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐻𝐻)
1027, 8, 45, 46, 47, 49, 100, 101ss2mcls 34162 . . . . . . 7 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st𝑥))(mCls‘𝑇)𝐻) ⊆ (𝑀(mCls‘𝑇)𝐻))
10389, 51eqeltrrd 2839 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽)
1043, 42, 45elmpps 34167 . . . . . . . . 9 (⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ∧ 𝐴 ∈ ((1st ‘(1st𝑥))(mCls‘𝑇)𝐻)))
105104simprbi 497 . . . . . . . 8 (⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽𝐴 ∈ ((1st ‘(1st𝑥))(mCls‘𝑇)𝐻))
106103, 105syl 17 . . . . . . 7 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐴 ∈ ((1st ‘(1st𝑥))(mCls‘𝑇)𝐻))
107102, 106sseldd 3945 . . . . . 6 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻))
10844, 107rexlimddv 3158 . . . . 5 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻))
1093, 42, 45elmpps 34167 . . . . 5 (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ∧ 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻)))
11040, 108, 109sylanbrc 583 . . . 4 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽)
1111ineq1i 4168 . . . . . . . 8 (𝑀 ∩ (𝑍 × 𝑍)) = ((𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ∩ (𝑍 × 𝑍))
112 indir 4235 . . . . . . . 8 ((𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ∩ (𝑍 × 𝑍)) = ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)))
113 disjdifr 4432 . . . . . . . . . 10 ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) = ∅
114 0ss 4356 . . . . . . . . . 10 ∅ ⊆ (𝐶 ∩ (𝑍 × 𝑍))
115113, 114eqsstri 3978 . . . . . . . . 9 ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) ⊆ (𝐶 ∩ (𝑍 × 𝑍))
116 ssequn2 4143 . . . . . . . . 9 (((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) ⊆ (𝐶 ∩ (𝑍 × 𝑍)) ↔ ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍))) = (𝐶 ∩ (𝑍 × 𝑍)))
117115, 116mpbi 229 . . . . . . . 8 ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍))) = (𝐶 ∩ (𝑍 × 𝑍))
118111, 112, 1173eqtri 2768 . . . . . . 7 (𝑀 ∩ (𝑍 × 𝑍)) = (𝐶 ∩ (𝑍 × 𝑍))
119118a1i 11 . . . . . 6 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑀 ∩ (𝑍 × 𝑍)) = (𝐶 ∩ (𝑍 × 𝑍)))
120119oteq1d 4842 . . . . 5 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨(𝑀 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
12159, 3, 41, 63msrval 34132 . . . . . 6 (⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) → (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = ⟨(𝑀 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
12240, 121syl 17 . . . . 5 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = ⟨(𝑀 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
123120, 122, 653eqtr4d 2786 . . . 4 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
124110, 123jca 512 . . 3 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩)))
125124ex 413 . 2 (𝑇 ∈ mFS → (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 → (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))))
12641, 42, 2mthmi 34171 . 2 ((⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩)) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈)
127125, 126impbid1 224 1 (𝑇 ∈ mFS → (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 ↔ (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wrex 3073  cdif 3907  cun 3908  cin 3909  wss 3910  c0 4282  {csn 4586  cotp 4594   cuni 4865   I cid 5530   × cxp 5631  ccnv 5632  cima 5636  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  Fincfn 8883  mVRcmvar 34055  mExcmex 34061  mDVcmdv 34062  mVarscmvrs 34063  mPreStcmpst 34067  mStRedcmsr 34068  mFScmfs 34070  mClscmcls 34071  mPPStcmpps 34072  mThmcmthm 34073
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 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
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 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-word 14403  df-concat 14459  df-s1 14484  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-gsum 17324  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-frmd 18659  df-mrex 34080  df-mex 34081  df-mdv 34082  df-mrsub 34084  df-msub 34085  df-mvh 34086  df-mpst 34087  df-msr 34088  df-msta 34089  df-mfs 34090  df-mcls 34091  df-mpps 34092  df-mthm 34093
This theorem is referenced by: (None)
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