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Theorem msubvrs 31653
Description: The set of variables in a substitution is the union, indexed by the variables in the original expression, of the variables in the substitution to that variable. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
msubvrs.s 𝑆 = (mSubst‘𝑇)
msubvrs.e 𝐸 = (mEx‘𝑇)
msubvrs.v 𝑉 = (mVars‘𝑇)
msubvrs.h 𝐻 = (mVH‘𝑇)
Assertion
Ref Expression
msubvrs ((𝑇 ∈ mFS ∧ 𝐹 ∈ ran 𝑆𝑋𝐸) → (𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))))
Distinct variable groups:   𝑥,𝐸   𝑥,𝐹   𝑥,𝑇   𝑥,𝑋   𝑥,𝑉
Allowed substitution hints:   𝑆(𝑥)   𝐻(𝑥)

Proof of Theorem msubvrs
Dummy variables 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 msubvrs.e . . . . . 6 𝐸 = (mEx‘𝑇)
2 eqid 2692 . . . . . 6 (mRSubst‘𝑇) = (mRSubst‘𝑇)
3 msubvrs.s . . . . . 6 𝑆 = (mSubst‘𝑇)
41, 2, 3elmsubrn 31621 . . . . 5 ran 𝑆 = ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
54eleq2i 2763 . . . 4 (𝐹 ∈ ran 𝑆𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
6 eqid 2692 . . . . 5 (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) = (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
7 fvex 6282 . . . . . . 7 (mEx‘𝑇) ∈ V
81, 7eqeltri 2767 . . . . . 6 𝐸 ∈ V
98mptex 6570 . . . . 5 (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) ∈ V
106, 9elrnmpti 5451 . . . 4 (𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
115, 10bitri 264 . . 3 (𝐹 ∈ ran 𝑆 ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
12 simp2 1129 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝑓 ∈ ran (mRSubst‘𝑇))
13 simp3 1130 . . . . . . . . . . 11 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝑋𝐸)
14 eqid 2692 . . . . . . . . . . . 12 (mTC‘𝑇) = (mTC‘𝑇)
15 eqid 2692 . . . . . . . . . . . 12 (mREx‘𝑇) = (mREx‘𝑇)
1614, 1, 15mexval 31595 . . . . . . . . . . 11 𝐸 = ((mTC‘𝑇) × (mREx‘𝑇))
1713, 16syl6eleq 2781 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝑋 ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
18 xp2nd 7286 . . . . . . . . . 10 (𝑋 ∈ ((mTC‘𝑇) × (mREx‘𝑇)) → (2nd𝑋) ∈ (mREx‘𝑇))
1917, 18syl 17 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (2nd𝑋) ∈ (mREx‘𝑇))
20 eqid 2692 . . . . . . . . . 10 (mVR‘𝑇) = (mVR‘𝑇)
212, 20, 15mrsubvrs 31615 . . . . . . . . 9 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ (2nd𝑋) ∈ (mREx‘𝑇)) → (ran (𝑓‘(2nd𝑋)) ∩ (mVR‘𝑇)) = 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))(ran (𝑓‘⟨“𝑥”⟩) ∩ (mVR‘𝑇)))
2212, 19, 21syl2anc 696 . . . . . . . 8 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (ran (𝑓‘(2nd𝑋)) ∩ (mVR‘𝑇)) = 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))(ran (𝑓‘⟨“𝑥”⟩) ∩ (mVR‘𝑇)))
23 fveq2 6272 . . . . . . . . . . . . 13 (𝑒 = 𝑋 → (1st𝑒) = (1st𝑋))
24 fveq2 6272 . . . . . . . . . . . . . 14 (𝑒 = 𝑋 → (2nd𝑒) = (2nd𝑋))
2524fveq2d 6276 . . . . . . . . . . . . 13 (𝑒 = 𝑋 → (𝑓‘(2nd𝑒)) = (𝑓‘(2nd𝑋)))
2623, 25opeq12d 4485 . . . . . . . . . . . 12 (𝑒 = 𝑋 → ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩ = ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩)
27 eqid 2692 . . . . . . . . . . . 12 (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)
28 opex 5005 . . . . . . . . . . . 12 ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩ ∈ V
2926, 27, 28fvmpt3i 6369 . . . . . . . . . . 11 (𝑋𝐸 → ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋) = ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩)
3013, 29syl 17 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋) = ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩)
3130fveq2d 6276 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋)) = (𝑉‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩))
32 xp1st 7285 . . . . . . . . . . . . 13 (𝑋 ∈ ((mTC‘𝑇) × (mREx‘𝑇)) → (1st𝑋) ∈ (mTC‘𝑇))
3317, 32syl 17 . . . . . . . . . . . 12 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (1st𝑋) ∈ (mTC‘𝑇))
342, 15mrsubf 31610 . . . . . . . . . . . . . 14 (𝑓 ∈ ran (mRSubst‘𝑇) → 𝑓:(mREx‘𝑇)⟶(mREx‘𝑇))
3512, 34syl 17 . . . . . . . . . . . . 13 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝑓:(mREx‘𝑇)⟶(mREx‘𝑇))
3618, 16eleq2s 2789 . . . . . . . . . . . . . 14 (𝑋𝐸 → (2nd𝑋) ∈ (mREx‘𝑇))
3713, 36syl 17 . . . . . . . . . . . . 13 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (2nd𝑋) ∈ (mREx‘𝑇))
3835, 37ffvelrnd 6443 . . . . . . . . . . . 12 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑓‘(2nd𝑋)) ∈ (mREx‘𝑇))
39 opelxpi 5225 . . . . . . . . . . . 12 (((1st𝑋) ∈ (mTC‘𝑇) ∧ (𝑓‘(2nd𝑋)) ∈ (mREx‘𝑇)) → ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
4033, 38, 39syl2anc 696 . . . . . . . . . . 11 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
4140, 16syl6eleqr 2782 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩ ∈ 𝐸)
42 msubvrs.v . . . . . . . . . . 11 𝑉 = (mVars‘𝑇)
4320, 1, 42mvrsval 31598 . . . . . . . . . 10 (⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩ ∈ 𝐸 → (𝑉‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) = (ran (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) ∩ (mVR‘𝑇)))
4441, 43syl 17 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) = (ran (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) ∩ (mVR‘𝑇)))
45 fvex 6282 . . . . . . . . . . . . 13 (1st𝑋) ∈ V
46 fvex 6282 . . . . . . . . . . . . 13 (𝑓‘(2nd𝑋)) ∈ V
4745, 46op2nd 7262 . . . . . . . . . . . 12 (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) = (𝑓‘(2nd𝑋))
4847a1i 11 . . . . . . . . . . 11 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) = (𝑓‘(2nd𝑋)))
4948rneqd 5428 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → ran (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) = ran (𝑓‘(2nd𝑋)))
5049ineq1d 3889 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (ran (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) ∩ (mVR‘𝑇)) = (ran (𝑓‘(2nd𝑋)) ∩ (mVR‘𝑇)))
5131, 44, 503eqtrd 2730 . . . . . . . 8 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋)) = (ran (𝑓‘(2nd𝑋)) ∩ (mVR‘𝑇)))
5220, 1, 42mvrsval 31598 . . . . . . . . . . 11 (𝑋𝐸 → (𝑉𝑋) = (ran (2nd𝑋) ∩ (mVR‘𝑇)))
5313, 52syl 17 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉𝑋) = (ran (2nd𝑋) ∩ (mVR‘𝑇)))
5453iuneq1d 4621 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝑥 ∈ (𝑉𝑋)(𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))) = 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))(𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))))
55 msubvrs.h . . . . . . . . . . . . . . . . 17 𝐻 = (mVH‘𝑇)
5620, 1, 55mvhf 31651 . . . . . . . . . . . . . . . 16 (𝑇 ∈ mFS → 𝐻:(mVR‘𝑇)⟶𝐸)
57563ad2ant1 1125 . . . . . . . . . . . . . . 15 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝐻:(mVR‘𝑇)⟶𝐸)
58 inss2 3910 . . . . . . . . . . . . . . . 16 (ran (2nd𝑋) ∩ (mVR‘𝑇)) ⊆ (mVR‘𝑇)
5958sseli 3673 . . . . . . . . . . . . . . 15 (𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇)) → 𝑥 ∈ (mVR‘𝑇))
60 ffvelrn 6440 . . . . . . . . . . . . . . 15 ((𝐻:(mVR‘𝑇)⟶𝐸𝑥 ∈ (mVR‘𝑇)) → (𝐻𝑥) ∈ 𝐸)
6157, 59, 60syl2an 495 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (𝐻𝑥) ∈ 𝐸)
62 fveq2 6272 . . . . . . . . . . . . . . . 16 (𝑒 = (𝐻𝑥) → (1st𝑒) = (1st ‘(𝐻𝑥)))
63 fveq2 6272 . . . . . . . . . . . . . . . . 17 (𝑒 = (𝐻𝑥) → (2nd𝑒) = (2nd ‘(𝐻𝑥)))
6463fveq2d 6276 . . . . . . . . . . . . . . . 16 (𝑒 = (𝐻𝑥) → (𝑓‘(2nd𝑒)) = (𝑓‘(2nd ‘(𝐻𝑥))))
6562, 64opeq12d 4485 . . . . . . . . . . . . . . 15 (𝑒 = (𝐻𝑥) → ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩ = ⟨(1st ‘(𝐻𝑥)), (𝑓‘(2nd ‘(𝐻𝑥)))⟩)
6665, 27, 28fvmpt3i 6369 . . . . . . . . . . . . . 14 ((𝐻𝑥) ∈ 𝐸 → ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥)) = ⟨(1st ‘(𝐻𝑥)), (𝑓‘(2nd ‘(𝐻𝑥)))⟩)
6761, 66syl 17 . . . . . . . . . . . . 13 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥)) = ⟨(1st ‘(𝐻𝑥)), (𝑓‘(2nd ‘(𝐻𝑥)))⟩)
6859adantl 473 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → 𝑥 ∈ (mVR‘𝑇))
69 eqid 2692 . . . . . . . . . . . . . . . . 17 (mType‘𝑇) = (mType‘𝑇)
7020, 69, 55mvhval 31627 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (mVR‘𝑇) → (𝐻𝑥) = ⟨((mType‘𝑇)‘𝑥), ⟨“𝑥”⟩⟩)
7168, 70syl 17 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (𝐻𝑥) = ⟨((mType‘𝑇)‘𝑥), ⟨“𝑥”⟩⟩)
72 fvex 6282 . . . . . . . . . . . . . . . 16 ((mType‘𝑇)‘𝑥) ∈ V
73 s1cli 13464 . . . . . . . . . . . . . . . . 17 ⟨“𝑥”⟩ ∈ Word V
7473elexi 3285 . . . . . . . . . . . . . . . 16 ⟨“𝑥”⟩ ∈ V
7572, 74op1std 7263 . . . . . . . . . . . . . . 15 ((𝐻𝑥) = ⟨((mType‘𝑇)‘𝑥), ⟨“𝑥”⟩⟩ → (1st ‘(𝐻𝑥)) = ((mType‘𝑇)‘𝑥))
7671, 75syl 17 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (1st ‘(𝐻𝑥)) = ((mType‘𝑇)‘𝑥))
7772, 74op2ndd 7264 . . . . . . . . . . . . . . . 16 ((𝐻𝑥) = ⟨((mType‘𝑇)‘𝑥), ⟨“𝑥”⟩⟩ → (2nd ‘(𝐻𝑥)) = ⟨“𝑥”⟩)
7871, 77syl 17 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (2nd ‘(𝐻𝑥)) = ⟨“𝑥”⟩)
7978fveq2d 6276 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (𝑓‘(2nd ‘(𝐻𝑥))) = (𝑓‘⟨“𝑥”⟩))
8076, 79opeq12d 4485 . . . . . . . . . . . . 13 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ⟨(1st ‘(𝐻𝑥)), (𝑓‘(2nd ‘(𝐻𝑥)))⟩ = ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩)
8167, 80eqtrd 2726 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥)) = ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩)
8281fveq2d 6276 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))) = (𝑉‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩))
83 simpl1 1146 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → 𝑇 ∈ mFS)
8420, 14, 69mtyf2 31644 . . . . . . . . . . . . . . . 16 (𝑇 ∈ mFS → (mType‘𝑇):(mVR‘𝑇)⟶(mTC‘𝑇))
8583, 84syl 17 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (mType‘𝑇):(mVR‘𝑇)⟶(mTC‘𝑇))
8685, 68ffvelrnd 6443 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ((mType‘𝑇)‘𝑥) ∈ (mTC‘𝑇))
8735adantr 472 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → 𝑓:(mREx‘𝑇)⟶(mREx‘𝑇))
88 elun2 3857 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (mVR‘𝑇) → 𝑥 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
8968, 88syl 17 . . . . . . . . . . . . . . . . 17 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → 𝑥 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
9089s1cld 13462 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ⟨“𝑥”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
91 eqid 2692 . . . . . . . . . . . . . . . . . 18 (mCN‘𝑇) = (mCN‘𝑇)
9291, 20, 15mrexval 31594 . . . . . . . . . . . . . . . . 17 (𝑇 ∈ mFS → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
9383, 92syl 17 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
9490, 93eleqtrrd 2774 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ⟨“𝑥”⟩ ∈ (mREx‘𝑇))
9587, 94ffvelrnd 6443 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (𝑓‘⟨“𝑥”⟩) ∈ (mREx‘𝑇))
96 opelxpi 5225 . . . . . . . . . . . . . 14 ((((mType‘𝑇)‘𝑥) ∈ (mTC‘𝑇) ∧ (𝑓‘⟨“𝑥”⟩) ∈ (mREx‘𝑇)) → ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
9786, 95, 96syl2anc 696 . . . . . . . . . . . . 13 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
9897, 16syl6eleqr 2782 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩ ∈ 𝐸)
9920, 1, 42mvrsval 31598 . . . . . . . . . . . 12 (⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩ ∈ 𝐸 → (𝑉‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) = (ran (2nd ‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) ∩ (mVR‘𝑇)))
10098, 99syl 17 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (𝑉‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) = (ran (2nd ‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) ∩ (mVR‘𝑇)))
101 fvex 6282 . . . . . . . . . . . . . . 15 (𝑓‘⟨“𝑥”⟩) ∈ V
10272, 101op2nd 7262 . . . . . . . . . . . . . 14 (2nd ‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) = (𝑓‘⟨“𝑥”⟩)
103102a1i 11 . . . . . . . . . . . . 13 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (2nd ‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) = (𝑓‘⟨“𝑥”⟩))
104103rneqd 5428 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ran (2nd ‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) = ran (𝑓‘⟨“𝑥”⟩))
105104ineq1d 3889 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (ran (2nd ‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) ∩ (mVR‘𝑇)) = (ran (𝑓‘⟨“𝑥”⟩) ∩ (mVR‘𝑇)))
10682, 100, 1053eqtrd 2730 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))) = (ran (𝑓‘⟨“𝑥”⟩) ∩ (mVR‘𝑇)))
107106iuneq2dv 4618 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))(𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))) = 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))(ran (𝑓‘⟨“𝑥”⟩) ∩ (mVR‘𝑇)))
10854, 107eqtrd 2726 . . . . . . . 8 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝑥 ∈ (𝑉𝑋)(𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))) = 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))(ran (𝑓‘⟨“𝑥”⟩) ∩ (mVR‘𝑇)))
10922, 51, 1083eqtr4d 2736 . . . . . . 7 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))))
110 fveq1 6271 . . . . . . . . 9 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝐹𝑋) = ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋))
111110fveq2d 6276 . . . . . . . 8 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑉‘(𝐹𝑋)) = (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋)))
112 fveq1 6271 . . . . . . . . . 10 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝐹‘(𝐻𝑥)) = ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥)))
113112fveq2d 6276 . . . . . . . . 9 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑉‘(𝐹‘(𝐻𝑥))) = (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))))
114113iuneq2d 4623 . . . . . . . 8 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))) = 𝑥 ∈ (𝑉𝑋)(𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))))
115111, 114eqeq12d 2707 . . . . . . 7 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → ((𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))) ↔ (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥)))))
116109, 115syl5ibrcom 237 . . . . . 6 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥)))))
1171163expia 1114 . . . . 5 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇)) → (𝑋𝐸 → (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))))))
118117com23 86 . . . 4 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇)) → (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑋𝐸 → (𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))))))
119118rexlimdva 3101 . . 3 (𝑇 ∈ mFS → (∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑋𝐸 → (𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))))))
12011, 119syl5bi 232 . 2 (𝑇 ∈ mFS → (𝐹 ∈ ran 𝑆 → (𝑋𝐸 → (𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))))))
1211203imp 1101 1 ((𝑇 ∈ mFS ∧ 𝐹 ∈ ran 𝑆𝑋𝐸) → (𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1564  wcel 2071  wrex 2983  Vcvv 3272  cun 3646  cin 3647  cop 4259   ciun 4596  cmpt 4805   × cxp 5184  ran crn 5187  wf 5965  cfv 5969  1st c1st 7251  2nd c2nd 7252  Word cword 13366  ⟨“cs1 13369  mCNcmcn 31553  mVRcmvar 31554  mTypecmty 31555  mTCcmtc 31557  mRExcmrex 31559  mExcmex 31560  mVarscmvrs 31562  mRSubstcmrsub 31563  mSubstcmsub 31564  mVHcmvh 31565  mFScmfs 31569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-8 2073  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-rep 4847  ax-sep 4857  ax-nul 4865  ax-pow 4916  ax-pr 4979  ax-un 7034  ax-cnex 10073  ax-resscn 10074  ax-1cn 10075  ax-icn 10076  ax-addcl 10077  ax-addrcl 10078  ax-mulcl 10079  ax-mulrcl 10080  ax-mulcom 10081  ax-addass 10082  ax-mulass 10083  ax-distr 10084  ax-i2m1 10085  ax-1ne0 10086  ax-1rid 10087  ax-rnegex 10088  ax-rrecex 10089  ax-cnre 10090  ax-pre-lttri 10091  ax-pre-lttrn 10092  ax-pre-ltadd 10093  ax-pre-mulgt0 10094
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-nel 2968  df-ral 2987  df-rex 2988  df-reu 2989  df-rmo 2990  df-rab 2991  df-v 3274  df-sbc 3510  df-csb 3608  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-pss 3664  df-nul 3992  df-if 4163  df-pw 4236  df-sn 4254  df-pr 4256  df-tp 4258  df-op 4260  df-uni 4513  df-int 4552  df-iun 4598  df-br 4729  df-opab 4789  df-mpt 4806  df-tr 4829  df-id 5096  df-eprel 5101  df-po 5107  df-so 5108  df-fr 5145  df-we 5147  df-xp 5192  df-rel 5193  df-cnv 5194  df-co 5195  df-dm 5196  df-rn 5197  df-res 5198  df-ima 5199  df-pred 5761  df-ord 5807  df-on 5808  df-lim 5809  df-suc 5810  df-iota 5932  df-fun 5971  df-fn 5972  df-f 5973  df-f1 5974  df-fo 5975  df-f1o 5976  df-fv 5977  df-riota 6694  df-ov 6736  df-oprab 6737  df-mpt2 6738  df-om 7151  df-1st 7253  df-2nd 7254  df-wrecs 7495  df-recs 7556  df-rdg 7594  df-1o 7648  df-oadd 7652  df-er 7830  df-map 7944  df-pm 7945  df-en 8041  df-dom 8042  df-sdom 8043  df-fin 8044  df-card 8846  df-pnf 10157  df-mnf 10158  df-xr 10159  df-ltxr 10160  df-le 10161  df-sub 10349  df-neg 10350  df-nn 11102  df-2 11160  df-n0 11374  df-xnn0 11445  df-z 11459  df-uz 11769  df-fz 12409  df-fzo 12549  df-seq 12885  df-hash 13201  df-word 13374  df-lsw 13375  df-concat 13376  df-s1 13377  df-substr 13378  df-struct 15950  df-ndx 15951  df-slot 15952  df-base 15954  df-sets 15955  df-ress 15956  df-plusg 16045  df-0g 16193  df-gsum 16194  df-mgm 17332  df-sgrp 17374  df-mnd 17385  df-submnd 17426  df-frmd 17476  df-mrex 31579  df-mex 31580  df-mvrs 31582  df-mrsub 31583  df-msub 31584  df-mvh 31585  df-mfs 31589
This theorem is referenced by:  mclsppslem  31676
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