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Theorem mthmpps 35410
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 2726 . . . . . . . . . . . . . 14 (mPreSt‘𝑇) = (mPreSt‘𝑇)
42, 3mthmsta 35406 . . . . . . . . . . . . 13 𝑈 ⊆ (mPreSt‘𝑇)
5 simpr 483 . . . . . . . . . . . . 13 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈)
64, 5sselid 3977 . . . . . . . . . . . 12 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇))
7 mthmpps.d . . . . . . . . . . . . 13 𝐷 = (mDV‘𝑇)
8 eqid 2726 . . . . . . . . . . . . 13 (mEx‘𝑇) = (mEx‘𝑇)
97, 8, 3elmpst 35364 . . . . . . . . . . . 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 1139 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶𝐷𝐶 = 𝐶))
1211simpld 493 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐶𝐷)
13 difssd 4132 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐷 ∖ (𝑍 × 𝑍)) ⊆ 𝐷)
1412, 13unssd 4187 . . . . . . . 8 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ⊆ 𝐷)
151, 14eqsstrid 4028 . . . . . . 7 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝑀𝐷)
1611simprd 494 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐶 = 𝐶)
17 cnvdif 6155 . . . . . . . . . . 11 (𝐷 ∖ (𝑍 × 𝑍)) = (𝐷(𝑍 × 𝑍))
18 cnvdif 6155 . . . . . . . . . . . . . 14 (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
19 cnvxp 6168 . . . . . . . . . . . . . . 15 ((mVR‘𝑇) × (mVR‘𝑇)) = ((mVR‘𝑇) × (mVR‘𝑇))
20 cnvi 6153 . . . . . . . . . . . . . . 15 I = I
2119, 20difeq12i 4119 . . . . . . . . . . . . . 14 (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
2218, 21eqtri 2754 . . . . . . . . . . . . 13 (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
23 eqid 2726 . . . . . . . . . . . . . . 15 (mVR‘𝑇) = (mVR‘𝑇)
2423, 7mdvval 35332 . . . . . . . . . . . . . 14 𝐷 = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
2524cnveqi 5881 . . . . . . . . . . . . 13 𝐷 = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
2622, 25, 243eqtr4i 2764 . . . . . . . . . . . 12 𝐷 = 𝐷
27 cnvxp 6168 . . . . . . . . . . . 12 (𝑍 × 𝑍) = (𝑍 × 𝑍)
2826, 27difeq12i 4119 . . . . . . . . . . 11 (𝐷(𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍))
2917, 28eqtri 2754 . . . . . . . . . 10 (𝐷 ∖ (𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍))
3029a1i 11 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐷 ∖ (𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍)))
3116, 30uneq12d 4164 . . . . . . . 8 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶(𝐷 ∖ (𝑍 × 𝑍))) = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))))
321cnveqi 5881 . . . . . . . . 9 𝑀 = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍)))
33 cnvun 6154 . . . . . . . . 9 (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) = (𝐶(𝐷 ∖ (𝑍 × 𝑍)))
3432, 33eqtri 2754 . . . . . . . 8 𝑀 = (𝐶(𝐷 ∖ (𝑍 × 𝑍)))
3531, 34, 13eqtr4g 2791 . . . . . . 7 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝑀 = 𝑀)
3615, 35jca 510 . . . . . 6 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑀𝐷𝑀 = 𝑀))
3710simp2d 1140 . . . . . 6 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin))
3810simp3d 1141 . . . . . 6 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐴 ∈ (mEx‘𝑇))
397, 8, 3elmpst 35364 . . . . . 6 (⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑀𝐷𝑀 = 𝑀) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
4036, 37, 38, 39syl3anbrc 1340 . . . . 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 35404 . . . . . . 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 2726 . . . . . . . 8 (mCls‘𝑇) = (mCls‘𝑇)
46 simpll 765 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑇 ∈ mFS)
4715adantr 479 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑀𝐷)
4837simpld 493 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐻 ⊆ (mEx‘𝑇))
4948adantr 479 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐻 ⊆ (mEx‘𝑇))
503, 42mppspst 35402 . . . . . . . . . . . . . . . . . . 19 𝐽 ⊆ (mPreSt‘𝑇)
51 simprl 769 . . . . . . . . . . . . . . . . . . 19 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥𝐽)
5250, 51sselid 3977 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 ∈ (mPreSt‘𝑇))
533mpst123 35368 . . . . . . . . . . . . . . . . . 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 6905 . . . . . . . . . . . . . . . 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 2768 . . . . . . . . . . . . . . 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 2827 . . . . . . . . . . . . . . . 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 2726 . . . . . . . . . . . . . . . . 17 (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) = (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)}))
6159, 3, 41, 60msrval 35366 . . . . . . . . . . . . . . . 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 35366 . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . 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 2774 . . . . . . . . . . . . . 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 6914 . . . . . . . . . . . . . . . 16 (1st ‘(1st𝑥)) ∈ V
6968inex1 5322 . . . . . . . . . . . . . . 15 ((1st ‘(1st𝑥)) ∩ ( (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) × (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})))) ∈ V
70 fvex 6914 . . . . . . . . . . . . . . 15 (2nd ‘(1st𝑥)) ∈ V
71 fvex 6914 . . . . . . . . . . . . . . 15 (2nd𝑥) ∈ V
7269, 70, 71otth 5490 . . . . . . . . . . . . . 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 1139 . . . . . . . . . . . 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 1140 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (2nd ‘(1st𝑥)) = 𝐻)
7673simp3d 1141 . . . . . . . . . . . . . . . . . . 19 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (2nd𝑥) = 𝐴)
7776sneqd 4645 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → {(2nd𝑥)} = {𝐴})
7875, 77uneq12d 4164 . . . . . . . . . . . . . . . . 17 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)}) = (𝐻 ∪ {𝐴}))
7978imaeq2d 6069 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) = (𝑉 “ (𝐻 ∪ {𝐴})))
8079unieqd 4926 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) = (𝑉 “ (𝐻 ∪ {𝐴})))
8180, 63eqtr4di 2784 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) = 𝑍)
8281sqxpeqd 5714 . . . . . . . . . . . . 13 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ( (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) × (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)}))) = (𝑍 × 𝑍))
8382ineq2d 4213 . . . . . . . . . . . 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 2768 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝐶 ∩ (𝑍 × 𝑍)) = ((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)))
85 inss1 4230 . . . . . . . . . . 11 (𝐶 ∩ (𝑍 × 𝑍)) ⊆ 𝐶
8684, 85eqsstrrdi 4035 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)) ⊆ 𝐶)
87 eqidd 2727 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (1st ‘(1st𝑥)) = (1st ‘(1st𝑥)))
8887, 75, 76oteq123d 4894 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1st ‘(1st𝑥)), (2nd ‘(1st𝑥)), (2nd𝑥)⟩ = ⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩)
8954, 88eqtrd 2766 . . . . . . . . . . . . 13 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 = ⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩)
9089, 52eqeltrrd 2827 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇))
917, 8, 3elmpst 35364 . . . . . . . . . . . . . 14 (⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ↔ (((1st ‘(1st𝑥)) ⊆ 𝐷(1st ‘(1st𝑥)) = (1st ‘(1st𝑥))) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
9291simp1bi 1142 . . . . . . . . . . . . 13 (⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) → ((1st ‘(1st𝑥)) ⊆ 𝐷(1st ‘(1st𝑥)) = (1st ‘(1st𝑥))))
9392simpld 493 . . . . . . . . . . . 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 4140 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st𝑥)) ∖ (𝑍 × 𝑍)) ⊆ (𝐷 ∖ (𝑍 × 𝑍)))
96 unss12 4183 . . . . . . . . . 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 582 . . . . . . . . 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 4483 . . . . . . . . . 10 (((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1st ‘(1st𝑥)) ∖ (𝑍 × 𝑍))) = (1st ‘(1st𝑥))
9998eqcomi 2735 . . . . . . . . 9 (1st ‘(1st𝑥)) = (((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1st ‘(1st𝑥)) ∖ (𝑍 × 𝑍)))
10097, 99, 13sstr4g 4025 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (1st ‘(1st𝑥)) ⊆ 𝑀)
101 ssidd 4003 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐻𝐻)
1027, 8, 45, 46, 47, 49, 100, 101ss2mcls 35396 . . . . . . 7 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st𝑥))(mCls‘𝑇)𝐻) ⊆ (𝑀(mCls‘𝑇)𝐻))
10389, 51eqeltrrd 2827 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽)
1043, 42, 45elmpps 35401 . . . . . . . . 9 (⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ∧ 𝐴 ∈ ((1st ‘(1st𝑥))(mCls‘𝑇)𝐻)))
105104simprbi 495 . . . . . . . 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 3980 . . . . . 6 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻))
10844, 107rexlimddv 3151 . . . . 5 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻))
1093, 42, 45elmpps 35401 . . . . 5 (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ∧ 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻)))
11040, 108, 109sylanbrc 581 . . . 4 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽)
1111ineq1i 4209 . . . . . . . 8 (𝑀 ∩ (𝑍 × 𝑍)) = ((𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ∩ (𝑍 × 𝑍))
112 indir 4277 . . . . . . . 8 ((𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ∩ (𝑍 × 𝑍)) = ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)))
113 disjdifr 4477 . . . . . . . . . 10 ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) = ∅
114 0ss 4401 . . . . . . . . . 10 ∅ ⊆ (𝐶 ∩ (𝑍 × 𝑍))
115113, 114eqsstri 4014 . . . . . . . . 9 ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) ⊆ (𝐶 ∩ (𝑍 × 𝑍))
116 ssequn2 4184 . . . . . . . . 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 2758 . . . . . . 7 (𝑀 ∩ (𝑍 × 𝑍)) = (𝐶 ∩ (𝑍 × 𝑍))
119118a1i 11 . . . . . 6 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑀 ∩ (𝑍 × 𝑍)) = (𝐶 ∩ (𝑍 × 𝑍)))
120119oteq1d 4891 . . . . 5 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨(𝑀 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
12159, 3, 41, 63msrval 35366 . . . . . 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 2776 . . . 4 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
124110, 123jca 510 . . 3 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩)))
125124ex 411 . 2 (𝑇 ∈ mFS → (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 → (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))))
12641, 42, 2mthmi 35405 . 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 394  w3a 1084   = wceq 1534  wcel 2099  wrex 3060  cdif 3944  cun 3945  cin 3946  wss 3947  c0 4325  {csn 4633  cotp 4641   cuni 4913   I cid 5579   × cxp 5680  ccnv 5681  cima 5685  cfv 6554  (class class class)co 7424  1st c1st 8001  2nd c2nd 8002  Fincfn 8974  mVRcmvar 35289  mExcmex 35295  mDVcmdv 35296  mVarscmvrs 35297  mPreStcmpst 35301  mStRedcmsr 35302  mFScmfs 35304  mClscmcls 35305  mPPStcmpps 35306  mThmcmthm 35307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-ot 4642  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-map 8857  df-pm 8858  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-n0 12525  df-z 12611  df-uz 12875  df-fz 13539  df-fzo 13682  df-seq 14022  df-hash 14348  df-word 14523  df-concat 14579  df-s1 14604  df-struct 17149  df-sets 17166  df-slot 17184  df-ndx 17196  df-base 17214  df-ress 17243  df-plusg 17279  df-0g 17456  df-gsum 17457  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-submnd 18774  df-frmd 18839  df-mrex 35314  df-mex 35315  df-mdv 35316  df-mrsub 35318  df-msub 35319  df-mvh 35320  df-mpst 35321  df-msr 35322  df-msta 35323  df-mfs 35324  df-mcls 35325  df-mpps 35326  df-mthm 35327
This theorem is referenced by: (None)
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