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Theorem mthmpps 31187
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 dv 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 2621 . . . . . . . . . . . . . 14 (mPreSt‘𝑇) = (mPreSt‘𝑇)
42, 3mthmsta 31183 . . . . . . . . . . . . 13 𝑈 ⊆ (mPreSt‘𝑇)
5 simpr 477 . . . . . . . . . . . . 13 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈)
64, 5sseldi 3581 . . . . . . . . . . . 12 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇))
7 mthmpps.d . . . . . . . . . . . . 13 𝐷 = (mDV‘𝑇)
8 eqid 2621 . . . . . . . . . . . . 13 (mEx‘𝑇) = (mEx‘𝑇)
97, 8, 3elmpst 31141 . . . . . . . . . . . 12 (⟨𝐶, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ↔ ((𝐶𝐷𝐶 = 𝐶) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
106, 9sylib 208 . . . . . . . . . . 11 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ((𝐶𝐷𝐶 = 𝐶) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
1110simp1d 1071 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶𝐷𝐶 = 𝐶))
1211simpld 475 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐶𝐷)
13 difssd 3716 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐷 ∖ (𝑍 × 𝑍)) ⊆ 𝐷)
1412, 13unssd 3767 . . . . . . . 8 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ⊆ 𝐷)
151, 14syl5eqss 3628 . . . . . . 7 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝑀𝐷)
1611simprd 479 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐶 = 𝐶)
17 cnvdif 5498 . . . . . . . . . . 11 (𝐷 ∖ (𝑍 × 𝑍)) = (𝐷(𝑍 × 𝑍))
18 cnvdif 5498 . . . . . . . . . . . . . 14 (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
19 cnvxp 5510 . . . . . . . . . . . . . . 15 ((mVR‘𝑇) × (mVR‘𝑇)) = ((mVR‘𝑇) × (mVR‘𝑇))
20 cnvi 5496 . . . . . . . . . . . . . . 15 I = I
2119, 20difeq12i 3704 . . . . . . . . . . . . . 14 (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
2218, 21eqtri 2643 . . . . . . . . . . . . 13 (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
23 eqid 2621 . . . . . . . . . . . . . . 15 (mVR‘𝑇) = (mVR‘𝑇)
2423, 7mdvval 31109 . . . . . . . . . . . . . 14 𝐷 = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
2524cnveqi 5257 . . . . . . . . . . . . 13 𝐷 = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
2622, 25, 243eqtr4i 2653 . . . . . . . . . . . 12 𝐷 = 𝐷
27 cnvxp 5510 . . . . . . . . . . . 12 (𝑍 × 𝑍) = (𝑍 × 𝑍)
2826, 27difeq12i 3704 . . . . . . . . . . 11 (𝐷(𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍))
2917, 28eqtri 2643 . . . . . . . . . 10 (𝐷 ∖ (𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍))
3029a1i 11 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐷 ∖ (𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍)))
3116, 30uneq12d 3746 . . . . . . . 8 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶(𝐷 ∖ (𝑍 × 𝑍))) = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))))
321cnveqi 5257 . . . . . . . . 9 𝑀 = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍)))
33 cnvun 5497 . . . . . . . . 9 (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) = (𝐶(𝐷 ∖ (𝑍 × 𝑍)))
3432, 33eqtri 2643 . . . . . . . 8 𝑀 = (𝐶(𝐷 ∖ (𝑍 × 𝑍)))
3531, 34, 13eqtr4g 2680 . . . . . . 7 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝑀 = 𝑀)
3615, 35jca 554 . . . . . 6 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑀𝐷𝑀 = 𝑀))
3710simp2d 1072 . . . . . 6 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin))
3810simp3d 1073 . . . . . 6 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐴 ∈ (mEx‘𝑇))
397, 8, 3elmpst 31141 . . . . . 6 (⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑀𝐷𝑀 = 𝑀) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
4036, 37, 38, 39syl3anbrc 1244 . . . . 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 31181 . . . . . . 7 (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
445, 43sylib 208 . . . . . 6 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ∃𝑥𝐽 (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
45 eqid 2621 . . . . . . . 8 (mCls‘𝑇) = (mCls‘𝑇)
46 simpll 789 . . . . . . . 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 475 . . . . . . . . 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 31179 . . . . . . . . . . . . . . . . . . 19 𝐽 ⊆ (mPreSt‘𝑇)
51 simprl 793 . . . . . . . . . . . . . . . . . . 19 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥𝐽)
5250, 51sseldi 3581 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 ∈ (mPreSt‘𝑇))
533mpst123 31145 . . . . . . . . . . . . . . . . . 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 6152 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅𝑥) = (𝑅‘⟨(1st ‘(1st𝑥)), (2nd ‘(1st𝑥)), (2nd𝑥)⟩))
56 simprr 795 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
5755, 56eqtr3d 2657 . . . . . . . . . . . . . . 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 2699 . . . . . . . . . . . . . . . 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 2621 . . . . . . . . . . . . . . . . 17 (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) = (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)}))
6159, 3, 41, 60msrval 31143 . . . . . . . . . . . . . . . 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 31143 . . . . . . . . . . . . . . . . 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 2663 . . . . . . . . . . . . . 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 6158 . . . . . . . . . . . . . . . 16 (1st ‘(1st𝑥)) ∈ V
6968inex1 4759 . . . . . . . . . . . . . . 15 ((1st ‘(1st𝑥)) ∩ ( (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) × (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})))) ∈ V
70 fvex 6158 . . . . . . . . . . . . . . 15 (2nd ‘(1st𝑥)) ∈ V
71 fvex 6158 . . . . . . . . . . . . . . 15 (2nd𝑥) ∈ V
7269, 70, 71otth 4913 . . . . . . . . . . . . . 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 208 . . . . . . . . . . . . 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 1071 . . . . . . . . . . . 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 1072 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (2nd ‘(1st𝑥)) = 𝐻)
7673simp3d 1073 . . . . . . . . . . . . . . . . . . 19 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (2nd𝑥) = 𝐴)
7776sneqd 4160 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → {(2nd𝑥)} = {𝐴})
7875, 77uneq12d 3746 . . . . . . . . . . . . . . . . 17 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)}) = (𝐻 ∪ {𝐴}))
7978imaeq2d 5425 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) = (𝑉 “ (𝐻 ∪ {𝐴})))
8079unieqd 4412 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) = (𝑉 “ (𝐻 ∪ {𝐴})))
8180, 63syl6eqr 2673 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) = 𝑍)
8281sqxpeqd 5101 . . . . . . . . . . . . 13 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ( (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) × (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)}))) = (𝑍 × 𝑍))
8382ineq2d 3792 . . . . . . . . . . . 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 2657 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝐶 ∩ (𝑍 × 𝑍)) = ((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)))
85 inss1 3811 . . . . . . . . . . 11 (𝐶 ∩ (𝑍 × 𝑍)) ⊆ 𝐶
8684, 85syl6eqssr 3635 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)) ⊆ 𝐶)
87 eqidd 2622 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (1st ‘(1st𝑥)) = (1st ‘(1st𝑥)))
8887, 75, 76oteq123d 4385 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1st ‘(1st𝑥)), (2nd ‘(1st𝑥)), (2nd𝑥)⟩ = ⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩)
8954, 88eqtrd 2655 . . . . . . . . . . . . 13 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 = ⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩)
9089, 52eqeltrrd 2699 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇))
917, 8, 3elmpst 31141 . . . . . . . . . . . . . 14 (⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ↔ (((1st ‘(1st𝑥)) ⊆ 𝐷(1st ‘(1st𝑥)) = (1st ‘(1st𝑥))) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
9291simp1bi 1074 . . . . . . . . . . . . 13 (⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) → ((1st ‘(1st𝑥)) ⊆ 𝐷(1st ‘(1st𝑥)) = (1st ‘(1st𝑥))))
9392simpld 475 . . . . . . . . . . . 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 3724 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st𝑥)) ∖ (𝑍 × 𝑍)) ⊆ (𝐷 ∖ (𝑍 × 𝑍)))
96 unss12 3763 . . . . . . . . . 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 692 . . . . . . . . 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 4018 . . . . . . . . . 10 (((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1st ‘(1st𝑥)) ∖ (𝑍 × 𝑍))) = (1st ‘(1st𝑥))
9998eqcomi 2630 . . . . . . . . 9 (1st ‘(1st𝑥)) = (((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1st ‘(1st𝑥)) ∖ (𝑍 × 𝑍)))
10097, 99, 13sstr4g 3625 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (1st ‘(1st𝑥)) ⊆ 𝑀)
101 ssid 3603 . . . . . . . . 9 𝐻𝐻
102101a1i 11 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐻𝐻)
1037, 8, 45, 46, 47, 49, 100, 102ss2mcls 31173 . . . . . . 7 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st𝑥))(mCls‘𝑇)𝐻) ⊆ (𝑀(mCls‘𝑇)𝐻))
10489, 51eqeltrrd 2699 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽)
1053, 42, 45elmpps 31178 . . . . . . . . 9 (⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ∧ 𝐴 ∈ ((1st ‘(1st𝑥))(mCls‘𝑇)𝐻)))
106105simprbi 480 . . . . . . . 8 (⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽𝐴 ∈ ((1st ‘(1st𝑥))(mCls‘𝑇)𝐻))
107104, 106syl 17 . . . . . . 7 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐴 ∈ ((1st ‘(1st𝑥))(mCls‘𝑇)𝐻))
108103, 107sseldd 3584 . . . . . 6 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻))
10944, 108rexlimddv 3028 . . . . 5 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻))
1103, 42, 45elmpps 31178 . . . . 5 (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ∧ 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻)))
11140, 109, 110sylanbrc 697 . . . 4 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽)
1121ineq1i 3788 . . . . . . . 8 (𝑀 ∩ (𝑍 × 𝑍)) = ((𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ∩ (𝑍 × 𝑍))
113 indir 3851 . . . . . . . 8 ((𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ∩ (𝑍 × 𝑍)) = ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)))
114 incom 3783 . . . . . . . . . . 11 ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) = ((𝑍 × 𝑍) ∩ (𝐷 ∖ (𝑍 × 𝑍)))
115 disjdif 4012 . . . . . . . . . . 11 ((𝑍 × 𝑍) ∩ (𝐷 ∖ (𝑍 × 𝑍))) = ∅
116114, 115eqtri 2643 . . . . . . . . . 10 ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) = ∅
117 0ss 3944 . . . . . . . . . 10 ∅ ⊆ (𝐶 ∩ (𝑍 × 𝑍))
118116, 117eqsstri 3614 . . . . . . . . 9 ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) ⊆ (𝐶 ∩ (𝑍 × 𝑍))
119 ssequn2 3764 . . . . . . . . 9 (((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) ⊆ (𝐶 ∩ (𝑍 × 𝑍)) ↔ ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍))) = (𝐶 ∩ (𝑍 × 𝑍)))
120118, 119mpbi 220 . . . . . . . 8 ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍))) = (𝐶 ∩ (𝑍 × 𝑍))
121112, 113, 1203eqtri 2647 . . . . . . 7 (𝑀 ∩ (𝑍 × 𝑍)) = (𝐶 ∩ (𝑍 × 𝑍))
122121a1i 11 . . . . . 6 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑀 ∩ (𝑍 × 𝑍)) = (𝐶 ∩ (𝑍 × 𝑍)))
123122oteq1d 4382 . . . . 5 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨(𝑀 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
12459, 3, 41, 63msrval 31143 . . . . . 6 (⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) → (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = ⟨(𝑀 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
12540, 124syl 17 . . . . 5 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = ⟨(𝑀 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
126123, 125, 653eqtr4d 2665 . . . 4 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
127111, 126jca 554 . . 3 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩)))
128127ex 450 . 2 (𝑇 ∈ mFS → (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 → (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))))
12941, 42, 2mthmi 31182 . 2 ((⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩)) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈)
130128, 129impbid1 215 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wrex 2908  cdif 3552  cun 3553  cin 3554  wss 3555  c0 3891  {csn 4148  cotp 4156   cuni 4402   I cid 4984   × cxp 5072  ccnv 5073  cima 5077  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  Fincfn 7899  mVRcmvar 31066  mExcmex 31072  mDVcmdv 31073  mVarscmvrs 31074  mPreStcmpst 31078  mStRedcmsr 31079  mFScmfs 31081  mClscmcls 31082  mPPStcmpps 31083  mThmcmthm 31084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-ot 4157  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-seq 12742  df-hash 13058  df-word 13238  df-concat 13240  df-s1 13241  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-0g 16023  df-gsum 16024  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-frmd 17307  df-mrex 31091  df-mex 31092  df-mdv 31093  df-mrsub 31095  df-msub 31096  df-mvh 31097  df-mpst 31098  df-msr 31099  df-msta 31100  df-mfs 31101  df-mcls 31102  df-mpps 31103  df-mthm 31104
This theorem is referenced by: (None)
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