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Theorem msubvrs 34154
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 2736 . . . . . 6 (mRSubst‘𝑇) = (mRSubst‘𝑇)
3 msubvrs.s . . . . . 6 𝑆 = (mSubst‘𝑇)
41, 2, 3elmsubrn 34122 . . . . 5 ran 𝑆 = ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
54eleq2i 2829 . . . 4 (𝐹 ∈ ran 𝑆𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
6 eqid 2736 . . . . 5 (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) = (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
71fvexi 6856 . . . . . 6 𝐸 ∈ V
87mptex 7173 . . . . 5 (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) ∈ V
96, 8elrnmpti 5915 . . . 4 (𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
105, 9bitri 274 . . 3 (𝐹 ∈ ran 𝑆 ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
11 simp2 1137 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝑓 ∈ ran (mRSubst‘𝑇))
12 simp3 1138 . . . . . . . . . . 11 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝑋𝐸)
13 eqid 2736 . . . . . . . . . . . 12 (mTC‘𝑇) = (mTC‘𝑇)
14 eqid 2736 . . . . . . . . . . . 12 (mREx‘𝑇) = (mREx‘𝑇)
1513, 1, 14mexval 34096 . . . . . . . . . . 11 𝐸 = ((mTC‘𝑇) × (mREx‘𝑇))
1612, 15eleqtrdi 2848 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝑋 ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
17 xp2nd 7954 . . . . . . . . . 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 2736 . . . . . . . . . 10 (mVR‘𝑇) = (mVR‘𝑇)
202, 19, 14mrsubvrs 34116 . . . . . . . . 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 584 . . . . . . . 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 6842 . . . . . . . . . . . . 13 (𝑒 = 𝑋 → (1st𝑒) = (1st𝑋))
23 2fveq3 6847 . . . . . . . . . . . . 13 (𝑒 = 𝑋 → (𝑓‘(2nd𝑒)) = (𝑓‘(2nd𝑋)))
2422, 23opeq12d 4838 . . . . . . . . . . . 12 (𝑒 = 𝑋 → ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩ = ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩)
25 eqid 2736 . . . . . . . . . . . 12 (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)
26 opex 5421 . . . . . . . . . . . 12 ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩ ∈ V
2724, 25, 26fvmpt3i 6953 . . . . . . . . . . 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 6846 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋)) = (𝑉‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩))
30 xp1st 7953 . . . . . . . . . . . . 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 34111 . . . . . . . . . . . . . 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 2856 . . . . . . . . . . . . . 14 (𝑋𝐸 → (2nd𝑋) ∈ (mREx‘𝑇))
3512, 34syl 17 . . . . . . . . . . . . 13 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (2nd𝑋) ∈ (mREx‘𝑇))
3633, 35ffvelcdmd 7036 . . . . . . . . . . . 12 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑓‘(2nd𝑋)) ∈ (mREx‘𝑇))
37 opelxpi 5670 . . . . . . . . . . . 12 (((1st𝑋) ∈ (mTC‘𝑇) ∧ (𝑓‘(2nd𝑋)) ∈ (mREx‘𝑇)) → ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
3831, 36, 37syl2anc 584 . . . . . . . . . . 11 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
3938, 15eleqtrrdi 2849 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩ ∈ 𝐸)
40 msubvrs.v . . . . . . . . . . 11 𝑉 = (mVars‘𝑇)
4119, 1, 40mvrsval 34099 . . . . . . . . . 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 6855 . . . . . . . . . . . . 13 (1st𝑋) ∈ V
44 fvex 6855 . . . . . . . . . . . . 13 (𝑓‘(2nd𝑋)) ∈ V
4543, 44op2nd 7930 . . . . . . . . . . . 12 (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) = (𝑓‘(2nd𝑋))
4645a1i 11 . . . . . . . . . . 11 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) = (𝑓‘(2nd𝑋)))
4746rneqd 5893 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → ran (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) = ran (𝑓‘(2nd𝑋)))
4847ineq1d 4171 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (ran (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) ∩ (mVR‘𝑇)) = (ran (𝑓‘(2nd𝑋)) ∩ (mVR‘𝑇)))
4929, 42, 483eqtrd 2780 . . . . . . . 8 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋)) = (ran (𝑓‘(2nd𝑋)) ∩ (mVR‘𝑇)))
5019, 1, 40mvrsval 34099 . . . . . . . . . . 11 (𝑋𝐸 → (𝑉𝑋) = (ran (2nd𝑋) ∩ (mVR‘𝑇)))
5112, 50syl 17 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉𝑋) = (ran (2nd𝑋) ∩ (mVR‘𝑇)))
5251iuneq1d 4981 . . . . . . . . 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 34152 . . . . . . . . . . . . . . . 16 (𝑇 ∈ mFS → 𝐻:(mVR‘𝑇)⟶𝐸)
55543ad2ant1 1133 . . . . . . . . . . . . . . 15 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝐻:(mVR‘𝑇)⟶𝐸)
56 inss2 4189 . . . . . . . . . . . . . . . 16 (ran (2nd𝑋) ∩ (mVR‘𝑇)) ⊆ (mVR‘𝑇)
5756sseli 3940 . . . . . . . . . . . . . . 15 (𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇)) → 𝑥 ∈ (mVR‘𝑇))
58 ffvelcdm 7032 . . . . . . . . . . . . . . 15 ((𝐻:(mVR‘𝑇)⟶𝐸𝑥 ∈ (mVR‘𝑇)) → (𝐻𝑥) ∈ 𝐸)
5955, 57, 58syl2an 596 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (𝐻𝑥) ∈ 𝐸)
60 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑒 = (𝐻𝑥) → (1st𝑒) = (1st ‘(𝐻𝑥)))
61 2fveq3 6847 . . . . . . . . . . . . . . . 16 (𝑒 = (𝐻𝑥) → (𝑓‘(2nd𝑒)) = (𝑓‘(2nd ‘(𝐻𝑥))))
6260, 61opeq12d 4838 . . . . . . . . . . . . . . 15 (𝑒 = (𝐻𝑥) → ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩ = ⟨(1st ‘(𝐻𝑥)), (𝑓‘(2nd ‘(𝐻𝑥)))⟩)
6362, 25, 26fvmpt3i 6953 . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → 𝑥 ∈ (mVR‘𝑇))
66 eqid 2736 . . . . . . . . . . . . . . . . 17 (mType‘𝑇) = (mType‘𝑇)
6719, 66, 53mvhval 34128 . . . . . . . . . . . . . . . 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 6855 . . . . . . . . . . . . . . . 16 ((mType‘𝑇)‘𝑥) ∈ V
70 s1cli 14493 . . . . . . . . . . . . . . . . 17 ⟨“𝑥”⟩ ∈ Word V
7170elexi 3464 . . . . . . . . . . . . . . . 16 ⟨“𝑥”⟩ ∈ V
7269, 71op1std 7931 . . . . . . . . . . . . . . 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 7932 . . . . . . . . . . . . . . . 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 6846 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (𝑓‘(2nd ‘(𝐻𝑥))) = (𝑓‘⟨“𝑥”⟩))
7773, 76opeq12d 4838 . . . . . . . . . . . . 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 2776 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥)) = ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩)
7978fveq2d 6846 . . . . . . . . . . 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 1191 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → 𝑇 ∈ mFS)
8119, 13, 66mtyf2 34145 . . . . . . . . . . . . . . . 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, 65ffvelcdmd 7036 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ((mType‘𝑇)‘𝑥) ∈ (mTC‘𝑇))
8433adantr 481 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → 𝑓:(mREx‘𝑇)⟶(mREx‘𝑇))
85 elun2 4137 . . . . . . . . . . . . . . . . . 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 14491 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ⟨“𝑥”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
88 eqid 2736 . . . . . . . . . . . . . . . . . 18 (mCN‘𝑇) = (mCN‘𝑇)
8988, 19, 14mrexval 34095 . . . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ⟨“𝑥”⟩ ∈ (mREx‘𝑇))
9284, 91ffvelcdmd 7036 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (𝑓‘⟨“𝑥”⟩) ∈ (mREx‘𝑇))
93 opelxpi 5670 . . . . . . . . . . . . . 14 ((((mType‘𝑇)‘𝑥) ∈ (mTC‘𝑇) ∧ (𝑓‘⟨“𝑥”⟩) ∈ (mREx‘𝑇)) → ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
9483, 92, 93syl2anc 584 . . . . . . . . . . . . 13 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
9594, 15eleqtrrdi 2849 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩ ∈ 𝐸)
9619, 1, 40mvrsval 34099 . . . . . . . . . . . 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 6855 . . . . . . . . . . . . . . 15 (𝑓‘⟨“𝑥”⟩) ∈ V
9969, 98op2nd 7930 . . . . . . . . . . . . . 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 5893 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ran (2nd ‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) = ran (𝑓‘⟨“𝑥”⟩))
102101ineq1d 4171 . . . . . . . . . . 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 2780 . . . . . . . . . 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 4978 . . . . . . . . 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 2776 . . . . . . . 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 2786 . . . . . . 7 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))))
107 fveq1 6841 . . . . . . . . 9 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝐹𝑋) = ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋))
108107fveq2d 6846 . . . . . . . 8 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑉‘(𝐹𝑋)) = (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋)))
109 fveq1 6841 . . . . . . . . . 10 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝐹‘(𝐻𝑥)) = ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥)))
110109fveq2d 6846 . . . . . . . . 9 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑉‘(𝐹‘(𝐻𝑥))) = (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))))
111110iuneq2d 4983 . . . . . . . 8 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))) = 𝑥 ∈ (𝑉𝑋)(𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))))
112108, 111eqeq12d 2752 . . . . . . 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 246 . . . . . 6 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥)))))
1141133expia 1121 . . . . 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 3152 . . 3 (𝑇 ∈ mFS → (∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑋𝐸 → (𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))))))
11710, 116biimtrid 241 . 2 (𝑇 ∈ mFS → (𝐹 ∈ ran 𝑆 → (𝑋𝐸 → (𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))))))
1181173imp 1111 1 ((𝑇 ∈ mFS ∧ 𝐹 ∈ ran 𝑆𝑋𝐸) → (𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wrex 3073  Vcvv 3445  cun 3908  cin 3909  cop 4592   ciun 4954  cmpt 5188   × cxp 5631  ran crn 5634  wf 6492  cfv 6496  1st c1st 7919  2nd c2nd 7920  Word cword 14402  ⟨“cs1 14483  mCNcmcn 34054  mVRcmvar 34055  mTypecmty 34056  mTCcmtc 34058  mRExcmrex 34060  mExcmex 34061  mVarscmvrs 34063  mRSubstcmrsub 34064  mSubstcmsub 34065  mVHcmvh 34066  mFScmfs 34070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-word 14403  df-lsw 14451  df-concat 14459  df-s1 14484  df-substr 14529  df-pfx 14559  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-gsum 17324  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-frmd 18659  df-mrex 34080  df-mex 34081  df-mvrs 34083  df-mrsub 34084  df-msub 34085  df-mvh 34086  df-mfs 34090
This theorem is referenced by:  mclsppslem  34177
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