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Theorem mthmpps 33107
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 2738 . . . . . . . . . . . . . 14 (mPreSt‘𝑇) = (mPreSt‘𝑇)
42, 3mthmsta 33103 . . . . . . . . . . . . 13 𝑈 ⊆ (mPreSt‘𝑇)
5 simpr 488 . . . . . . . . . . . . 13 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈)
64, 5sseldi 3873 . . . . . . . . . . . 12 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇))
7 mthmpps.d . . . . . . . . . . . . 13 𝐷 = (mDV‘𝑇)
8 eqid 2738 . . . . . . . . . . . . 13 (mEx‘𝑇) = (mEx‘𝑇)
97, 8, 3elmpst 33061 . . . . . . . . . . . 12 (⟨𝐶, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ↔ ((𝐶𝐷𝐶 = 𝐶) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
106, 9sylib 221 . . . . . . . . . . 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 498 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐶𝐷)
13 difssd 4021 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐷 ∖ (𝑍 × 𝑍)) ⊆ 𝐷)
1412, 13unssd 4074 . . . . . . . 8 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ⊆ 𝐷)
151, 14eqsstrid 3923 . . . . . . 7 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝑀𝐷)
1611simprd 499 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐶 = 𝐶)
17 cnvdif 5970 . . . . . . . . . . 11 (𝐷 ∖ (𝑍 × 𝑍)) = (𝐷(𝑍 × 𝑍))
18 cnvdif 5970 . . . . . . . . . . . . . 14 (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
19 cnvxp 5983 . . . . . . . . . . . . . . 15 ((mVR‘𝑇) × (mVR‘𝑇)) = ((mVR‘𝑇) × (mVR‘𝑇))
20 cnvi 5968 . . . . . . . . . . . . . . 15 I = I
2119, 20difeq12i 4009 . . . . . . . . . . . . . 14 (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
2218, 21eqtri 2761 . . . . . . . . . . . . 13 (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
23 eqid 2738 . . . . . . . . . . . . . . 15 (mVR‘𝑇) = (mVR‘𝑇)
2423, 7mdvval 33029 . . . . . . . . . . . . . 14 𝐷 = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
2524cnveqi 5711 . . . . . . . . . . . . 13 𝐷 = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
2622, 25, 243eqtr4i 2771 . . . . . . . . . . . 12 𝐷 = 𝐷
27 cnvxp 5983 . . . . . . . . . . . 12 (𝑍 × 𝑍) = (𝑍 × 𝑍)
2826, 27difeq12i 4009 . . . . . . . . . . 11 (𝐷(𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍))
2917, 28eqtri 2761 . . . . . . . . . 10 (𝐷 ∖ (𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍))
3029a1i 11 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐷 ∖ (𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍)))
3116, 30uneq12d 4052 . . . . . . . 8 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶(𝐷 ∖ (𝑍 × 𝑍))) = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))))
321cnveqi 5711 . . . . . . . . 9 𝑀 = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍)))
33 cnvun 5969 . . . . . . . . 9 (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) = (𝐶(𝐷 ∖ (𝑍 × 𝑍)))
3432, 33eqtri 2761 . . . . . . . 8 𝑀 = (𝐶(𝐷 ∖ (𝑍 × 𝑍)))
3531, 34, 13eqtr4g 2798 . . . . . . 7 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝑀 = 𝑀)
3615, 35jca 515 . . . . . 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 33061 . . . . . 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 33101 . . . . . . 7 (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
445, 43sylib 221 . . . . . 6 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ∃𝑥𝐽 (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
45 eqid 2738 . . . . . . . 8 (mCls‘𝑇) = (mCls‘𝑇)
46 simpll 767 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑇 ∈ mFS)
4715adantr 484 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑀𝐷)
4837simpld 498 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐻 ⊆ (mEx‘𝑇))
4948adantr 484 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐻 ⊆ (mEx‘𝑇))
503, 42mppspst 33099 . . . . . . . . . . . . . . . . . . 19 𝐽 ⊆ (mPreSt‘𝑇)
51 simprl 771 . . . . . . . . . . . . . . . . . . 19 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥𝐽)
5250, 51sseldi 3873 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 ∈ (mPreSt‘𝑇))
533mpst123 33065 . . . . . . . . . . . . . . . . . 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 6672 . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . 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 2834 . . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . . . 17 (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) = (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)}))
6159, 3, 41, 60msrval 33063 . . . . . . . . . . . . . . . 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 33063 . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . 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 2781 . . . . . . . . . . . . . 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 6681 . . . . . . . . . . . . . . . 16 (1st ‘(1st𝑥)) ∈ V
6968inex1 5182 . . . . . . . . . . . . . . 15 ((1st ‘(1st𝑥)) ∩ ( (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) × (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})))) ∈ V
70 fvex 6681 . . . . . . . . . . . . . . 15 (2nd ‘(1st𝑥)) ∈ V
71 fvex 6681 . . . . . . . . . . . . . . 15 (2nd𝑥) ∈ V
7269, 70, 71otth 5339 . . . . . . . . . . . . . 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 221 . . . . . . . . . . . . 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 4525 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → {(2nd𝑥)} = {𝐴})
7875, 77uneq12d 4052 . . . . . . . . . . . . . . . . 17 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)}) = (𝐻 ∪ {𝐴}))
7978imaeq2d 5897 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) = (𝑉 “ (𝐻 ∪ {𝐴})))
8079unieqd 4807 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) = (𝑉 “ (𝐻 ∪ {𝐴})))
8180, 63eqtr4di 2791 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) = 𝑍)
8281sqxpeqd 5551 . . . . . . . . . . . . 13 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ( (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)})) × (𝑉 “ ((2nd ‘(1st𝑥)) ∪ {(2nd𝑥)}))) = (𝑍 × 𝑍))
8382ineq2d 4101 . . . . . . . . . . . 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 2775 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝐶 ∩ (𝑍 × 𝑍)) = ((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)))
85 inss1 4117 . . . . . . . . . . 11 (𝐶 ∩ (𝑍 × 𝑍)) ⊆ 𝐶
8684, 85eqsstrrdi 3930 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)) ⊆ 𝐶)
87 eqidd 2739 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (1st ‘(1st𝑥)) = (1st ‘(1st𝑥)))
8887, 75, 76oteq123d 4773 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1st ‘(1st𝑥)), (2nd ‘(1st𝑥)), (2nd𝑥)⟩ = ⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩)
8954, 88eqtrd 2773 . . . . . . . . . . . . 13 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 = ⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩)
9089, 52eqeltrrd 2834 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇))
917, 8, 3elmpst 33061 . . . . . . . . . . . . . 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 498 . . . . . . . . . . . 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 4029 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st𝑥)) ∖ (𝑍 × 𝑍)) ⊆ (𝐷 ∖ (𝑍 × 𝑍)))
96 unss12 4070 . . . . . . . . . 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 587 . . . . . . . . 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 4365 . . . . . . . . . 10 (((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1st ‘(1st𝑥)) ∖ (𝑍 × 𝑍))) = (1st ‘(1st𝑥))
9998eqcomi 2747 . . . . . . . . 9 (1st ‘(1st𝑥)) = (((1st ‘(1st𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1st ‘(1st𝑥)) ∖ (𝑍 × 𝑍)))
10097, 99, 13sstr4g 3920 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (1st ‘(1st𝑥)) ⊆ 𝑀)
101 ssidd 3898 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐻𝐻)
1027, 8, 45, 46, 47, 49, 100, 101ss2mcls 33093 . . . . . . 7 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st𝑥))(mCls‘𝑇)𝐻) ⊆ (𝑀(mCls‘𝑇)𝐻))
10389, 51eqeltrrd 2834 . . . . . . . 8 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽)
1043, 42, 45elmpps 33098 . . . . . . . . 9 (⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨(1st ‘(1st𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ∧ 𝐴 ∈ ((1st ‘(1st𝑥))(mCls‘𝑇)𝐻)))
105104simprbi 500 . . . . . . . 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 3876 . . . . . 6 (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥𝐽 ∧ (𝑅𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻))
10844, 107rexlimddv 3200 . . . . 5 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻))
1093, 42, 45elmpps 33098 . . . . 5 (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ∧ 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻)))
11040, 108, 109sylanbrc 586 . . . 4 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽)
1111ineq1i 4097 . . . . . . . 8 (𝑀 ∩ (𝑍 × 𝑍)) = ((𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ∩ (𝑍 × 𝑍))
112 indir 4164 . . . . . . . 8 ((𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ∩ (𝑍 × 𝑍)) = ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)))
113 disjdifr 4359 . . . . . . . . . 10 ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) = ∅
114 0ss 4282 . . . . . . . . . 10 ∅ ⊆ (𝐶 ∩ (𝑍 × 𝑍))
115113, 114eqsstri 3909 . . . . . . . . 9 ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) ⊆ (𝐶 ∩ (𝑍 × 𝑍))
116 ssequn2 4071 . . . . . . . . 9 (((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) ⊆ (𝐶 ∩ (𝑍 × 𝑍)) ↔ ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍))) = (𝐶 ∩ (𝑍 × 𝑍)))
117115, 116mpbi 233 . . . . . . . 8 ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍))) = (𝐶 ∩ (𝑍 × 𝑍))
118111, 112, 1173eqtri 2765 . . . . . . 7 (𝑀 ∩ (𝑍 × 𝑍)) = (𝐶 ∩ (𝑍 × 𝑍))
119118a1i 11 . . . . . 6 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑀 ∩ (𝑍 × 𝑍)) = (𝐶 ∩ (𝑍 × 𝑍)))
120119oteq1d 4770 . . . . 5 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨(𝑀 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
12159, 3, 41, 63msrval 33063 . . . . . 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 2783 . . . 4 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
124110, 123jca 515 . . 3 ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩)))
125124ex 416 . 2 (𝑇 ∈ mFS → (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 → (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))))
12641, 42, 2mthmi 33102 . 2 ((⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩)) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈)
127125, 126impbid1 228 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 209  wa 399  w3a 1088   = wceq 1542  wcel 2113  wrex 3054  cdif 3838  cun 3839  cin 3840  wss 3841  c0 4209  {csn 4513  cotp 4521   cuni 4793   I cid 5424   × cxp 5517  ccnv 5518  cima 5522  cfv 6333  (class class class)co 7164  1st c1st 7705  2nd c2nd 7706  Fincfn 8548  mVRcmvar 32986  mExcmex 32992  mDVcmdv 32993  mVarscmvrs 32994  mPreStcmpst 32998  mStRedcmsr 32999  mFScmfs 33001  mClscmcls 33002  mPPStcmpps 33003  mThmcmthm 33004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473  ax-cnex 10664  ax-resscn 10665  ax-1cn 10666  ax-icn 10667  ax-addcl 10668  ax-addrcl 10669  ax-mulcl 10670  ax-mulrcl 10671  ax-mulcom 10672  ax-addass 10673  ax-mulass 10674  ax-distr 10675  ax-i2m1 10676  ax-1ne0 10677  ax-1rid 10678  ax-rnegex 10679  ax-rrecex 10680  ax-cnre 10681  ax-pre-lttri 10682  ax-pre-lttrn 10683  ax-pre-ltadd 10684  ax-pre-mulgt0 10685
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-ot 4522  df-uni 4794  df-int 4834  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-1st 7707  df-2nd 7708  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-1o 8124  df-er 8313  df-map 8432  df-pm 8433  df-en 8549  df-dom 8550  df-sdom 8551  df-fin 8552  df-card 9434  df-pnf 10748  df-mnf 10749  df-xr 10750  df-ltxr 10751  df-le 10752  df-sub 10943  df-neg 10944  df-nn 11710  df-2 11772  df-n0 11970  df-z 12056  df-uz 12318  df-fz 12975  df-fzo 13118  df-seq 13454  df-hash 13776  df-word 13949  df-concat 14005  df-s1 14032  df-struct 16581  df-ndx 16582  df-slot 16583  df-base 16585  df-sets 16586  df-ress 16587  df-plusg 16674  df-0g 16811  df-gsum 16812  df-mgm 17961  df-sgrp 18010  df-mnd 18021  df-submnd 18066  df-frmd 18123  df-mrex 33011  df-mex 33012  df-mdv 33013  df-mrsub 33015  df-msub 33016  df-mvh 33017  df-mpst 33018  df-msr 33019  df-msta 33020  df-mfs 33021  df-mcls 33022  df-mpps 33023  df-mthm 33024
This theorem is referenced by: (None)
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