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Theorem mthmpps 35587
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 2737 . . . . . . . . . . . . . 14 (mPreSt‘𝑇) = (mPreSt‘𝑇)
42, 3mthmsta 35583 . . . . . . . . . . . . 13 𝑈 ⊆ (mPreSt‘𝑇)
5 simpr 484 . . . . . . . . . . . . 13 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈)
64, 5sselid 3981 . . . . . . . . . . . 12 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇))
7 mthmpps.d . . . . . . . . . . . . 13 𝐷 = (mDV‘𝑇)
8 eqid 2737 . . . . . . . . . . . . 13 (mEx‘𝑇) = (mEx‘𝑇)
97, 8, 3elmpst 35541 . . . . . . . . . . . 12 (⟨𝐶, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ↔ ((𝐶𝐷𝐶 = 𝐶) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
106, 9sylib 218 . . . . . . . . . . 11 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ((𝐶𝐷𝐶 = 𝐶) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
1110simp1d 1143 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶𝐷𝐶 = 𝐶))
1211simpld 494 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐶𝐷)
13 difssd 4137 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐷 ∖ (𝑍 × 𝑍)) ⊆ 𝐷)
1412, 13unssd 4192 . . . . . . . 8 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ⊆ 𝐷)
151, 14eqsstrid 4022 . . . . . . 7 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝑀𝐷)
1611simprd 495 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐶 = 𝐶)
17 cnvdif 6163 . . . . . . . . . . 11 (𝐷 ∖ (𝑍 × 𝑍)) = (𝐷(𝑍 × 𝑍))
18 cnvdif 6163 . . . . . . . . . . . . . 14 (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
19 cnvxp 6177 . . . . . . . . . . . . . . 15 ((mVR‘𝑇) × (mVR‘𝑇)) = ((mVR‘𝑇) × (mVR‘𝑇))
20 cnvi 6161 . . . . . . . . . . . . . . 15 I = I
2119, 20difeq12i 4124 . . . . . . . . . . . . . 14 (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
2218, 21eqtri 2765 . . . . . . . . . . . . 13 (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
23 eqid 2737 . . . . . . . . . . . . . . 15 (mVR‘𝑇) = (mVR‘𝑇)
2423, 7mdvval 35509 . . . . . . . . . . . . . 14 𝐷 = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
2524cnveqi 5885 . . . . . . . . . . . . 13 𝐷 = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
2622, 25, 243eqtr4i 2775 . . . . . . . . . . . 12 𝐷 = 𝐷
27 cnvxp 6177 . . . . . . . . . . . 12 (𝑍 × 𝑍) = (𝑍 × 𝑍)
2826, 27difeq12i 4124 . . . . . . . . . . 11 (𝐷(𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍))
2917, 28eqtri 2765 . . . . . . . . . 10 (𝐷 ∖ (𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍))
3029a1i 11 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐷 ∖ (𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍)))
3116, 30uneq12d 4169 . . . . . . . 8 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶(𝐷 ∖ (𝑍 × 𝑍))) = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))))
321cnveqi 5885 . . . . . . . . 9 𝑀 = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍)))
33 cnvun 6162 . . . . . . . . 9 (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) = (𝐶(𝐷 ∖ (𝑍 × 𝑍)))
3432, 33eqtri 2765 . . . . . . . 8 𝑀 = (𝐶(𝐷 ∖ (𝑍 × 𝑍)))
3531, 34, 13eqtr4g 2802 . . . . . . 7 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝑀 = 𝑀)
3615, 35jca 511 . . . . . 6 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑀𝐷𝑀 = 𝑀))
3710simp2d 1144 . . . . . 6 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin))
3810simp3d 1145 . . . . . 6 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐴 ∈ (mEx‘𝑇))
397, 8, 3elmpst 35541 . . . . . 6 (⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑀𝐷𝑀 = 𝑀) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
4036, 37, 38, 39syl3anbrc 1344 . . . . 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 35581 . . . . . . 7 (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
445, 43sylib 218 . . . . . 6 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ∃𝑥𝐽 (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
45 eqid 2737 . . . . . . . 8 (mCls‘𝑇) = (mCls‘𝑇)
46 simpll 767 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑇 ∈ mFS)
4715adantr 480 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑀𝐷)
4837simpld 494 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐻 ⊆ (mEx‘𝑇))
4948adantr 480 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐻 ⊆ (mEx‘𝑇))
503, 42mppspst 35579 . . . . . . . . . . . . . . . . . . 19 𝐽 ⊆ (mPreSt‘𝑇)
51 simprl 771 . . . . . . . . . . . . . . . . . . 19 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥𝐽)
5250, 51sselid 3981 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 ∈ (mPreSt‘𝑇))
533mpst123 35545 . . . . . . . . . . . . . . . . . 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 6910 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅𝑥) = (𝑅‘⟨(1st ‘(1st𝑥)), (2nd ‘(1st𝑥)), (2nd𝑥)⟩))
56 simprr 773 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
5755, 56eqtr3d 2779 . . . . . . . . . . . . . . 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 2842 . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . 17 (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) = (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)}))
6159, 3, 41, 60msrval 35543 . . . . . . . . . . . . . . . 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 35543 . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 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 2785 . . . . . . . . . . . . . 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 6919 . . . . . . . . . . . . . . . 16 (1st ‘(1st𝑥)) ∈ V
6968inex1 5317 . . . . . . . . . . . . . . 15 ((1st ‘(1st𝑥)) ∩ ( (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) × (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})))) ∈ V
70 fvex 6919 . . . . . . . . . . . . . . 15 (2nd ‘(1st𝑥)) ∈ V
71 fvex 6919 . . . . . . . . . . . . . . 15 (2nd𝑥) ∈ V
7269, 70, 71otth 5489 . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . 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 1143 . . . . . . . . . . . 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 1144 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (2nd ‘(1st𝑥)) = 𝐻)
7673simp3d 1145 . . . . . . . . . . . . . . . . . . 19 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (2nd𝑥) = 𝐴)
7776sneqd 4638 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → {(2nd𝑥)} = {𝐴})
7875, 77uneq12d 4169 . . . . . . . . . . . . . . . . 17 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)}) = (𝐻 ∪ {𝐴}))
7978imaeq2d 6078 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) = (𝑉 “ (𝐻 ∪ {𝐴})))
8079unieqd 4920 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) = (𝑉 “ (𝐻 ∪ {𝐴})))
8180, 63eqtr4di 2795 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) = 𝑍)
8281sqxpeqd 5717 . . . . . . . . . . . . 13 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ( (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) × (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)}))) = (𝑍 × 𝑍))
8382ineq2d 4220 . . . . . . . . . . . 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 2779 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝐶 ∩ (𝑍 × 𝑍)) = ((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)))
85 inss1 4237 . . . . . . . . . . 11 (𝐶 ∩ (𝑍 × 𝑍)) ⊆ 𝐶
8684, 85eqsstrrdi 4029 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)) ⊆ 𝐶)
87 eqidd 2738 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (1st ‘(1st𝑥)) = (1st ‘(1st𝑥)))
8887, 75, 76oteq123d 4888 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1st ‘(1st𝑥)), (2nd ‘(1st𝑥)), (2nd𝑥)⟩ = ⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩)
8954, 88eqtrd 2777 . . . . . . . . . . . . 13 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 = ⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩)
9089, 52eqeltrrd 2842 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇))
917, 8, 3elmpst 35541 . . . . . . . . . . . . . 14 (⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ↔ (((1st ‘(1st𝑥)) ⊆ 𝐷(1st ‘(1st𝑥)) = (1st ‘(1st𝑥))) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
9291simp1bi 1146 . . . . . . . . . . . . 13 (⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) → ((1st ‘(1st𝑥)) ⊆ 𝐷(1st ‘(1st𝑥)) = (1st ‘(1st𝑥))))
9392simpld 494 . . . . . . . . . . . 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 4145 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st𝑥)) ∖ (𝑍 × 𝑍)) ⊆ (𝐷 ∖ (𝑍 × 𝑍)))
96 unss12 4188 . . . . . . . . . 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 4479 . . . . . . . . . 10 (((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1st ‘(1st𝑥)) ∖ (𝑍 × 𝑍))) = (1st ‘(1st𝑥))
9998eqcomi 2746 . . . . . . . . 9 (1st ‘(1st𝑥)) = (((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1st ‘(1st𝑥)) ∖ (𝑍 × 𝑍)))
10097, 99, 13sstr4g 4037 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (1st ‘(1st𝑥)) ⊆ 𝑀)
101 ssidd 4007 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐻𝐻)
1027, 8, 45, 46, 47, 49, 100, 101ss2mcls 35573 . . . . . . 7 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st𝑥))(mCls‘𝑇)𝐻) ⊆ (𝑀(mCls‘𝑇)𝐻))
10389, 51eqeltrrd 2842 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽)
1043, 42, 45elmpps 35578 . . . . . . . . 9 (⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ∧ 𝐴 ∈ ((1st ‘(1st𝑥))(mCls‘𝑇)𝐻)))
105104simprbi 496 . . . . . . . 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 3984 . . . . . 6 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻))
10844, 107rexlimddv 3161 . . . . 5 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻))
1093, 42, 45elmpps 35578 . . . . 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 4216 . . . . . . . 8 (𝑀 ∩ (𝑍 × 𝑍)) = ((𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ∩ (𝑍 × 𝑍))
112 indir 4286 . . . . . . . 8 ((𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ∩ (𝑍 × 𝑍)) = ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)))
113 disjdifr 4473 . . . . . . . . . 10 ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) = ∅
114 0ss 4400 . . . . . . . . . 10 ∅ ⊆ (𝐶 ∩ (𝑍 × 𝑍))
115113, 114eqsstri 4030 . . . . . . . . 9 ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) ⊆ (𝐶 ∩ (𝑍 × 𝑍))
116 ssequn2 4189 . . . . . . . . 9 (((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) ⊆ (𝐶 ∩ (𝑍 × 𝑍)) ↔ ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍))) = (𝐶 ∩ (𝑍 × 𝑍)))
117115, 116mpbi 230 . . . . . . . 8 ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍))) = (𝐶 ∩ (𝑍 × 𝑍))
118111, 112, 1173eqtri 2769 . . . . . . 7 (𝑀 ∩ (𝑍 × 𝑍)) = (𝐶 ∩ (𝑍 × 𝑍))
119118a1i 11 . . . . . 6 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑀 ∩ (𝑍 × 𝑍)) = (𝐶 ∩ (𝑍 × 𝑍)))
120119oteq1d 4885 . . . . 5 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨(𝑀 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
12159, 3, 41, 63msrval 35543 . . . . . 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 2787 . . . 4 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
124110, 123jca 511 . . 3 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩)))
125124ex 412 . 2 (𝑇 ∈ mFS → (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 → (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))))
12641, 42, 2mthmi 35582 . 2 ((⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩)) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈)
127125, 126impbid1 225 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wrex 3070  cdif 3948  cun 3949  cin 3950  wss 3951  c0 4333  {csn 4626  cotp 4634   cuni 4907   I cid 5577   × cxp 5683  ccnv 5684  cima 5688  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  Fincfn 8985  mVRcmvar 35466  mExcmex 35472  mDVcmdv 35473  mVarscmvrs 35474  mPreStcmpst 35478  mStRedcmsr 35479  mFScmfs 35481  mClscmcls 35482  mPPStcmpps 35483  mThmcmthm 35484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-word 14553  df-concat 14609  df-s1 14634  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-gsum 17487  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-frmd 18862  df-mrex 35491  df-mex 35492  df-mdv 35493  df-mrsub 35495  df-msub 35496  df-mvh 35497  df-mpst 35498  df-msr 35499  df-msta 35500  df-mfs 35501  df-mcls 35502  df-mpps 35503  df-mthm 35504
This theorem is referenced by: (None)
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