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Theorem msubvrs 35565
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 2737 . . . . . 6 (mRSubst‘𝑇) = (mRSubst‘𝑇)
3 msubvrs.s . . . . . 6 𝑆 = (mSubst‘𝑇)
41, 2, 3elmsubrn 35533 . . . . 5 ran 𝑆 = ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
54eleq2i 2833 . . . 4 (𝐹 ∈ ran 𝑆𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
6 eqid 2737 . . . . 5 (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) = (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
71fvexi 6920 . . . . . 6 𝐸 ∈ V
87mptex 7243 . . . . 5 (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) ∈ V
96, 8elrnmpti 5973 . . . 4 (𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
105, 9bitri 275 . . 3 (𝐹 ∈ ran 𝑆 ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
11 simp2 1138 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝑓 ∈ ran (mRSubst‘𝑇))
12 simp3 1139 . . . . . . . . . . 11 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝑋𝐸)
13 eqid 2737 . . . . . . . . . . . 12 (mTC‘𝑇) = (mTC‘𝑇)
14 eqid 2737 . . . . . . . . . . . 12 (mREx‘𝑇) = (mREx‘𝑇)
1513, 1, 14mexval 35507 . . . . . . . . . . 11 𝐸 = ((mTC‘𝑇) × (mREx‘𝑇))
1612, 15eleqtrdi 2851 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝑋 ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
17 xp2nd 8047 . . . . . . . . . 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 2737 . . . . . . . . . 10 (mVR‘𝑇) = (mVR‘𝑇)
202, 19, 14mrsubvrs 35527 . . . . . . . . 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 6906 . . . . . . . . . . . . 13 (𝑒 = 𝑋 → (1st𝑒) = (1st𝑋))
23 2fveq3 6911 . . . . . . . . . . . . 13 (𝑒 = 𝑋 → (𝑓‘(2nd𝑒)) = (𝑓‘(2nd𝑋)))
2422, 23opeq12d 4881 . . . . . . . . . . . 12 (𝑒 = 𝑋 → ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩ = ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩)
25 eqid 2737 . . . . . . . . . . . 12 (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)
26 opex 5469 . . . . . . . . . . . 12 ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩ ∈ V
2724, 25, 26fvmpt3i 7021 . . . . . . . . . . 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 6910 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋)) = (𝑉‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩))
30 xp1st 8046 . . . . . . . . . . . . 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 35522 . . . . . . . . . . . . . 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 2859 . . . . . . . . . . . . . 14 (𝑋𝐸 → (2nd𝑋) ∈ (mREx‘𝑇))
3512, 34syl 17 . . . . . . . . . . . . 13 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (2nd𝑋) ∈ (mREx‘𝑇))
3633, 35ffvelcdmd 7105 . . . . . . . . . . . 12 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑓‘(2nd𝑋)) ∈ (mREx‘𝑇))
37 opelxpi 5722 . . . . . . . . . . . 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 2852 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩ ∈ 𝐸)
40 msubvrs.v . . . . . . . . . . 11 𝑉 = (mVars‘𝑇)
4119, 1, 40mvrsval 35510 . . . . . . . . . 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 6919 . . . . . . . . . . . . 13 (1st𝑋) ∈ V
44 fvex 6919 . . . . . . . . . . . . 13 (𝑓‘(2nd𝑋)) ∈ V
4543, 44op2nd 8023 . . . . . . . . . . . 12 (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) = (𝑓‘(2nd𝑋))
4645a1i 11 . . . . . . . . . . 11 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) = (𝑓‘(2nd𝑋)))
4746rneqd 5949 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → ran (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) = ran (𝑓‘(2nd𝑋)))
4847ineq1d 4219 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (ran (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) ∩ (mVR‘𝑇)) = (ran (𝑓‘(2nd𝑋)) ∩ (mVR‘𝑇)))
4929, 42, 483eqtrd 2781 . . . . . . . 8 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋)) = (ran (𝑓‘(2nd𝑋)) ∩ (mVR‘𝑇)))
5019, 1, 40mvrsval 35510 . . . . . . . . . . 11 (𝑋𝐸 → (𝑉𝑋) = (ran (2nd𝑋) ∩ (mVR‘𝑇)))
5112, 50syl 17 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉𝑋) = (ran (2nd𝑋) ∩ (mVR‘𝑇)))
5251iuneq1d 5019 . . . . . . . . 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 35563 . . . . . . . . . . . . . . . 16 (𝑇 ∈ mFS → 𝐻:(mVR‘𝑇)⟶𝐸)
55543ad2ant1 1134 . . . . . . . . . . . . . . 15 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝐻:(mVR‘𝑇)⟶𝐸)
56 inss2 4238 . . . . . . . . . . . . . . . 16 (ran (2nd𝑋) ∩ (mVR‘𝑇)) ⊆ (mVR‘𝑇)
5756sseli 3979 . . . . . . . . . . . . . . 15 (𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇)) → 𝑥 ∈ (mVR‘𝑇))
58 ffvelcdm 7101 . . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . 16 (𝑒 = (𝐻𝑥) → (1st𝑒) = (1st ‘(𝐻𝑥)))
61 2fveq3 6911 . . . . . . . . . . . . . . . 16 (𝑒 = (𝐻𝑥) → (𝑓‘(2nd𝑒)) = (𝑓‘(2nd ‘(𝐻𝑥))))
6260, 61opeq12d 4881 . . . . . . . . . . . . . . 15 (𝑒 = (𝐻𝑥) → ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩ = ⟨(1st ‘(𝐻𝑥)), (𝑓‘(2nd ‘(𝐻𝑥)))⟩)
6362, 25, 26fvmpt3i 7021 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → 𝑥 ∈ (mVR‘𝑇))
66 eqid 2737 . . . . . . . . . . . . . . . . 17 (mType‘𝑇) = (mType‘𝑇)
6719, 66, 53mvhval 35539 . . . . . . . . . . . . . . . 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 6919 . . . . . . . . . . . . . . . 16 ((mType‘𝑇)‘𝑥) ∈ V
70 s1cli 14643 . . . . . . . . . . . . . . . . 17 ⟨“𝑥”⟩ ∈ Word V
7170elexi 3503 . . . . . . . . . . . . . . . 16 ⟨“𝑥”⟩ ∈ V
7269, 71op1std 8024 . . . . . . . . . . . . . . 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 8025 . . . . . . . . . . . . . . . 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 6910 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (𝑓‘(2nd ‘(𝐻𝑥))) = (𝑓‘⟨“𝑥”⟩))
7773, 76opeq12d 4881 . . . . . . . . . . . . 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 2777 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥)) = ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩)
7978fveq2d 6910 . . . . . . . . . . 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 1192 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → 𝑇 ∈ mFS)
8119, 13, 66mtyf2 35556 . . . . . . . . . . . . . . . 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 7105 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ((mType‘𝑇)‘𝑥) ∈ (mTC‘𝑇))
8433adantr 480 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → 𝑓:(mREx‘𝑇)⟶(mREx‘𝑇))
85 elun2 4183 . . . . . . . . . . . . . . . . . 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 14641 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ⟨“𝑥”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
88 eqid 2737 . . . . . . . . . . . . . . . . . 18 (mCN‘𝑇) = (mCN‘𝑇)
8988, 19, 14mrexval 35506 . . . . . . . . . . . . . . . . 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 2844 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ⟨“𝑥”⟩ ∈ (mREx‘𝑇))
9284, 91ffvelcdmd 7105 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (𝑓‘⟨“𝑥”⟩) ∈ (mREx‘𝑇))
93 opelxpi 5722 . . . . . . . . . . . . . 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 2852 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩ ∈ 𝐸)
9619, 1, 40mvrsval 35510 . . . . . . . . . . . 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 6919 . . . . . . . . . . . . . . 15 (𝑓‘⟨“𝑥”⟩) ∈ V
9969, 98op2nd 8023 . . . . . . . . . . . . . 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 5949 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ran (2nd ‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) = ran (𝑓‘⟨“𝑥”⟩))
102101ineq1d 4219 . . . . . . . . . . 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 2781 . . . . . . . . . 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 5016 . . . . . . . . 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 2777 . . . . . . . 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 2787 . . . . . . 7 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))))
107 fveq1 6905 . . . . . . . . 9 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝐹𝑋) = ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋))
108107fveq2d 6910 . . . . . . . 8 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑉‘(𝐹𝑋)) = (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋)))
109 fveq1 6905 . . . . . . . . . 10 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝐹‘(𝐻𝑥)) = ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥)))
110109fveq2d 6910 . . . . . . . . 9 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑉‘(𝐹‘(𝐻𝑥))) = (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))))
111110iuneq2d 5022 . . . . . . . 8 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))) = 𝑥 ∈ (𝑉𝑋)(𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))))
112108, 111eqeq12d 2753 . . . . . . 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 247 . . . . . 6 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥)))))
1141133expia 1122 . . . . 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 3155 . . 3 (𝑇 ∈ mFS → (∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑋𝐸 → (𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))))))
11710, 116biimtrid 242 . 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 395  w3a 1087   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  cun 3949  cin 3950  cop 4632   ciun 4991  cmpt 5225   × cxp 5683  ran crn 5686  wf 6557  cfv 6561  1st c1st 8012  2nd c2nd 8013  Word cword 14552  ⟨“cs1 14633  mCNcmcn 35465  mVRcmvar 35466  mTypecmty 35467  mTCcmtc 35469  mRExcmrex 35471  mExcmex 35472  mVarscmvrs 35474  mRSubstcmrsub 35475  mSubstcmsub 35476  mVHcmvh 35477  mFScmfs 35481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-word 14553  df-lsw 14601  df-concat 14609  df-s1 14634  df-substr 14679  df-pfx 14709  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-gsum 17487  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-frmd 18862  df-mrex 35491  df-mex 35492  df-mvrs 35494  df-mrsub 35495  df-msub 35496  df-mvh 35497  df-mfs 35501
This theorem is referenced by:  mclsppslem  35588
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