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Theorem msubvrs 32056
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 2777 . . . . . 6 (mRSubst‘𝑇) = (mRSubst‘𝑇)
3 msubvrs.s . . . . . 6 𝑆 = (mSubst‘𝑇)
41, 2, 3elmsubrn 32024 . . . . 5 ran 𝑆 = ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
54eleq2i 2850 . . . 4 (𝐹 ∈ ran 𝑆𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
6 eqid 2777 . . . . 5 (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) = (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
71fvexi 6460 . . . . . 6 𝐸 ∈ V
87mptex 6758 . . . . 5 (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) ∈ V
96, 8elrnmpti 5622 . . . 4 (𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
105, 9bitri 267 . . 3 (𝐹 ∈ ran 𝑆 ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
11 simp2 1128 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝑓 ∈ ran (mRSubst‘𝑇))
12 simp3 1129 . . . . . . . . . . 11 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝑋𝐸)
13 eqid 2777 . . . . . . . . . . . 12 (mTC‘𝑇) = (mTC‘𝑇)
14 eqid 2777 . . . . . . . . . . . 12 (mREx‘𝑇) = (mREx‘𝑇)
1513, 1, 14mexval 31998 . . . . . . . . . . 11 𝐸 = ((mTC‘𝑇) × (mREx‘𝑇))
1612, 15syl6eleq 2868 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝑋 ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
17 xp2nd 7478 . . . . . . . . . 10 (𝑋 ∈ ((mTC‘𝑇) × (mREx‘𝑇)) → (2nd𝑋) ∈ (mREx‘𝑇))
1816, 17syl 17 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (2nd𝑋) ∈ (mREx‘𝑇))
19 eqid 2777 . . . . . . . . . 10 (mVR‘𝑇) = (mVR‘𝑇)
202, 19, 14mrsubvrs 32018 . . . . . . . . 9 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ (2nd𝑋) ∈ (mREx‘𝑇)) → (ran (𝑓‘(2nd𝑋)) ∩ (mVR‘𝑇)) = 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))(ran (𝑓‘⟨“𝑥”⟩) ∩ (mVR‘𝑇)))
2111, 18, 20syl2anc 579 . . . . . . . 8 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (ran (𝑓‘(2nd𝑋)) ∩ (mVR‘𝑇)) = 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))(ran (𝑓‘⟨“𝑥”⟩) ∩ (mVR‘𝑇)))
22 fveq2 6446 . . . . . . . . . . . . 13 (𝑒 = 𝑋 → (1st𝑒) = (1st𝑋))
23 2fveq3 6451 . . . . . . . . . . . . 13 (𝑒 = 𝑋 → (𝑓‘(2nd𝑒)) = (𝑓‘(2nd𝑋)))
2422, 23opeq12d 4644 . . . . . . . . . . . 12 (𝑒 = 𝑋 → ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩ = ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩)
25 eqid 2777 . . . . . . . . . . . 12 (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)
26 opex 5164 . . . . . . . . . . . 12 ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩ ∈ V
2724, 25, 26fvmpt3i 6547 . . . . . . . . . . 11 (𝑋𝐸 → ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋) = ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩)
2812, 27syl 17 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋) = ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩)
2928fveq2d 6450 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋)) = (𝑉‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩))
30 xp1st 7477 . . . . . . . . . . . . 13 (𝑋 ∈ ((mTC‘𝑇) × (mREx‘𝑇)) → (1st𝑋) ∈ (mTC‘𝑇))
3116, 30syl 17 . . . . . . . . . . . 12 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (1st𝑋) ∈ (mTC‘𝑇))
322, 14mrsubf 32013 . . . . . . . . . . . . . 14 (𝑓 ∈ ran (mRSubst‘𝑇) → 𝑓:(mREx‘𝑇)⟶(mREx‘𝑇))
3311, 32syl 17 . . . . . . . . . . . . 13 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝑓:(mREx‘𝑇)⟶(mREx‘𝑇))
3417, 15eleq2s 2876 . . . . . . . . . . . . . 14 (𝑋𝐸 → (2nd𝑋) ∈ (mREx‘𝑇))
3512, 34syl 17 . . . . . . . . . . . . 13 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (2nd𝑋) ∈ (mREx‘𝑇))
3633, 35ffvelrnd 6624 . . . . . . . . . . . 12 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑓‘(2nd𝑋)) ∈ (mREx‘𝑇))
37 opelxpi 5392 . . . . . . . . . . . 12 (((1st𝑋) ∈ (mTC‘𝑇) ∧ (𝑓‘(2nd𝑋)) ∈ (mREx‘𝑇)) → ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
3831, 36, 37syl2anc 579 . . . . . . . . . . 11 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
3938, 15syl6eleqr 2869 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩ ∈ 𝐸)
40 msubvrs.v . . . . . . . . . . 11 𝑉 = (mVars‘𝑇)
4119, 1, 40mvrsval 32001 . . . . . . . . . 10 (⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩ ∈ 𝐸 → (𝑉‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) = (ran (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) ∩ (mVR‘𝑇)))
4239, 41syl 17 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) = (ran (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) ∩ (mVR‘𝑇)))
43 fvex 6459 . . . . . . . . . . . . 13 (1st𝑋) ∈ V
44 fvex 6459 . . . . . . . . . . . . 13 (𝑓‘(2nd𝑋)) ∈ V
4543, 44op2nd 7454 . . . . . . . . . . . 12 (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) = (𝑓‘(2nd𝑋))
4645a1i 11 . . . . . . . . . . 11 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) = (𝑓‘(2nd𝑋)))
4746rneqd 5598 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → ran (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) = ran (𝑓‘(2nd𝑋)))
4847ineq1d 4035 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (ran (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) ∩ (mVR‘𝑇)) = (ran (𝑓‘(2nd𝑋)) ∩ (mVR‘𝑇)))
4929, 42, 483eqtrd 2817 . . . . . . . 8 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋)) = (ran (𝑓‘(2nd𝑋)) ∩ (mVR‘𝑇)))
5019, 1, 40mvrsval 32001 . . . . . . . . . . 11 (𝑋𝐸 → (𝑉𝑋) = (ran (2nd𝑋) ∩ (mVR‘𝑇)))
5112, 50syl 17 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉𝑋) = (ran (2nd𝑋) ∩ (mVR‘𝑇)))
5251iuneq1d 4778 . . . . . . . . 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))))
53 msubvrs.h . . . . . . . . . . . . . . . . 17 𝐻 = (mVH‘𝑇)
5419, 1, 53mvhf 32054 . . . . . . . . . . . . . . . 16 (𝑇 ∈ mFS → 𝐻:(mVR‘𝑇)⟶𝐸)
55543ad2ant1 1124 . . . . . . . . . . . . . . 15 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝐻:(mVR‘𝑇)⟶𝐸)
56 inss2 4053 . . . . . . . . . . . . . . . 16 (ran (2nd𝑋) ∩ (mVR‘𝑇)) ⊆ (mVR‘𝑇)
5756sseli 3816 . . . . . . . . . . . . . . 15 (𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇)) → 𝑥 ∈ (mVR‘𝑇))
58 ffvelrn 6621 . . . . . . . . . . . . . . 15 ((𝐻:(mVR‘𝑇)⟶𝐸𝑥 ∈ (mVR‘𝑇)) → (𝐻𝑥) ∈ 𝐸)
5955, 57, 58syl2an 589 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (𝐻𝑥) ∈ 𝐸)
60 fveq2 6446 . . . . . . . . . . . . . . . 16 (𝑒 = (𝐻𝑥) → (1st𝑒) = (1st ‘(𝐻𝑥)))
61 2fveq3 6451 . . . . . . . . . . . . . . . 16 (𝑒 = (𝐻𝑥) → (𝑓‘(2nd𝑒)) = (𝑓‘(2nd ‘(𝐻𝑥))))
6260, 61opeq12d 4644 . . . . . . . . . . . . . . 15 (𝑒 = (𝐻𝑥) → ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩ = ⟨(1st ‘(𝐻𝑥)), (𝑓‘(2nd ‘(𝐻𝑥)))⟩)
6362, 25, 26fvmpt3i 6547 . . . . . . . . . . . . . 14 ((𝐻𝑥) ∈ 𝐸 → ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥)) = ⟨(1st ‘(𝐻𝑥)), (𝑓‘(2nd ‘(𝐻𝑥)))⟩)
6459, 63syl 17 . . . . . . . . . . . . 13 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥)) = ⟨(1st ‘(𝐻𝑥)), (𝑓‘(2nd ‘(𝐻𝑥)))⟩)
6557adantl 475 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → 𝑥 ∈ (mVR‘𝑇))
66 eqid 2777 . . . . . . . . . . . . . . . . 17 (mType‘𝑇) = (mType‘𝑇)
6719, 66, 53mvhval 32030 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (mVR‘𝑇) → (𝐻𝑥) = ⟨((mType‘𝑇)‘𝑥), ⟨“𝑥”⟩⟩)
6865, 67syl 17 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (𝐻𝑥) = ⟨((mType‘𝑇)‘𝑥), ⟨“𝑥”⟩⟩)
69 fvex 6459 . . . . . . . . . . . . . . . 16 ((mType‘𝑇)‘𝑥) ∈ V
70 s1cli 13695 . . . . . . . . . . . . . . . . 17 ⟨“𝑥”⟩ ∈ Word V
7170elexi 3414 . . . . . . . . . . . . . . . 16 ⟨“𝑥”⟩ ∈ V
7269, 71op1std 7455 . . . . . . . . . . . . . . 15 ((𝐻𝑥) = ⟨((mType‘𝑇)‘𝑥), ⟨“𝑥”⟩⟩ → (1st ‘(𝐻𝑥)) = ((mType‘𝑇)‘𝑥))
7368, 72syl 17 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (1st ‘(𝐻𝑥)) = ((mType‘𝑇)‘𝑥))
7469, 71op2ndd 7456 . . . . . . . . . . . . . . . 16 ((𝐻𝑥) = ⟨((mType‘𝑇)‘𝑥), ⟨“𝑥”⟩⟩ → (2nd ‘(𝐻𝑥)) = ⟨“𝑥”⟩)
7568, 74syl 17 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (2nd ‘(𝐻𝑥)) = ⟨“𝑥”⟩)
7675fveq2d 6450 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (𝑓‘(2nd ‘(𝐻𝑥))) = (𝑓‘⟨“𝑥”⟩))
7773, 76opeq12d 4644 . . . . . . . . . . . . 13 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ⟨(1st ‘(𝐻𝑥)), (𝑓‘(2nd ‘(𝐻𝑥)))⟩ = ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩)
7864, 77eqtrd 2813 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥)) = ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩)
7978fveq2d 6450 . . . . . . . . . . 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”⟩)⟩))
80 simpl1 1199 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → 𝑇 ∈ mFS)
8119, 13, 66mtyf2 32047 . . . . . . . . . . . . . . . 16 (𝑇 ∈ mFS → (mType‘𝑇):(mVR‘𝑇)⟶(mTC‘𝑇))
8280, 81syl 17 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (mType‘𝑇):(mVR‘𝑇)⟶(mTC‘𝑇))
8382, 65ffvelrnd 6624 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ((mType‘𝑇)‘𝑥) ∈ (mTC‘𝑇))
8433adantr 474 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → 𝑓:(mREx‘𝑇)⟶(mREx‘𝑇))
85 elun2 4003 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (mVR‘𝑇) → 𝑥 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
8665, 85syl 17 . . . . . . . . . . . . . . . . 17 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → 𝑥 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
8786s1cld 13693 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ⟨“𝑥”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
88 eqid 2777 . . . . . . . . . . . . . . . . . 18 (mCN‘𝑇) = (mCN‘𝑇)
8988, 19, 14mrexval 31997 . . . . . . . . . . . . . . . . 17 (𝑇 ∈ mFS → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
9080, 89syl 17 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
9187, 90eleqtrrd 2861 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ⟨“𝑥”⟩ ∈ (mREx‘𝑇))
9284, 91ffvelrnd 6624 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (𝑓‘⟨“𝑥”⟩) ∈ (mREx‘𝑇))
93 opelxpi 5392 . . . . . . . . . . . . . 14 ((((mType‘𝑇)‘𝑥) ∈ (mTC‘𝑇) ∧ (𝑓‘⟨“𝑥”⟩) ∈ (mREx‘𝑇)) → ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
9483, 92, 93syl2anc 579 . . . . . . . . . . . . 13 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
9594, 15syl6eleqr 2869 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩ ∈ 𝐸)
9619, 1, 40mvrsval 32001 . . . . . . . . . . . 12 (⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩ ∈ 𝐸 → (𝑉‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) = (ran (2nd ‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) ∩ (mVR‘𝑇)))
9795, 96syl 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)))
98 fvex 6459 . . . . . . . . . . . . . . 15 (𝑓‘⟨“𝑥”⟩) ∈ V
9969, 98op2nd 7454 . . . . . . . . . . . . . 14 (2nd ‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) = (𝑓‘⟨“𝑥”⟩)
10099a1i 11 . . . . . . . . . . . . 13 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (2nd ‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) = (𝑓‘⟨“𝑥”⟩))
101100rneqd 5598 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ran (2nd ‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) = ran (𝑓‘⟨“𝑥”⟩))
102101ineq1d 4035 . . . . . . . . . . 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)))
10379, 97, 1023eqtrd 2817 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))) = (ran (𝑓‘⟨“𝑥”⟩) ∩ (mVR‘𝑇)))
104103iuneq2dv 4775 . . . . . . . . 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)))
10552, 104eqtrd 2813 . . . . . . . 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)))
10621, 49, 1053eqtr4d 2823 . . . . . . 7 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))))
107 fveq1 6445 . . . . . . . . 9 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝐹𝑋) = ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋))
108107fveq2d 6450 . . . . . . . 8 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑉‘(𝐹𝑋)) = (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋)))
109 fveq1 6445 . . . . . . . . . 10 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝐹‘(𝐻𝑥)) = ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥)))
110109fveq2d 6450 . . . . . . . . 9 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑉‘(𝐹‘(𝐻𝑥))) = (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))))
111110iuneq2d 4780 . . . . . . . 8 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))) = 𝑥 ∈ (𝑉𝑋)(𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))))
112108, 111eqeq12d 2792 . . . . . . 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)))))
113106, 112syl5ibrcom 239 . . . . . 6 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥)))))
1141133expia 1111 . . . . 5 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇)) → (𝑋𝐸 → (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))))))
115114com23 86 . . . 4 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇)) → (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑋𝐸 → (𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))))))
116115rexlimdva 3212 . . 3 (𝑇 ∈ mFS → (∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑋𝐸 → (𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))))))
11710, 116syl5bi 234 . 2 (𝑇 ∈ mFS → (𝐹 ∈ ran 𝑆 → (𝑋𝐸 → (𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))))))
1181173imp 1098 1 ((𝑇 ∈ mFS ∧ 𝐹 ∈ ran 𝑆𝑋𝐸) → (𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2106  wrex 3090  Vcvv 3397  cun 3789  cin 3790  cop 4403   ciun 4753  cmpt 4965   × cxp 5353  ran crn 5356  wf 6131  cfv 6135  1st c1st 7443  2nd c2nd 7444  Word cword 13599  ⟨“cs1 13685  mCNcmcn 31956  mVRcmvar 31957  mTypecmty 31958  mTCcmtc 31960  mRExcmrex 31962  mExcmex 31963  mVarscmvrs 31965  mRSubstcmrsub 31966  mSubstcmsub 31967  mVHcmvh 31968  mFScmfs 31972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-n0 11643  df-xnn0 11715  df-z 11729  df-uz 11993  df-fz 12644  df-fzo 12785  df-seq 13120  df-hash 13436  df-word 13600  df-lsw 13653  df-concat 13661  df-s1 13686  df-substr 13731  df-pfx 13780  df-struct 16257  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-ress 16263  df-plusg 16351  df-0g 16488  df-gsum 16489  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-submnd 17722  df-frmd 17773  df-mrex 31982  df-mex 31983  df-mvrs 31985  df-mrsub 31986  df-msub 31987  df-mvh 31988  df-mfs 31992
This theorem is referenced by:  mclsppslem  32079
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