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Theorem msubvrs 31583
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 2651 . . . . . 6 (mRSubst‘𝑇) = (mRSubst‘𝑇)
3 msubvrs.s . . . . . 6 𝑆 = (mSubst‘𝑇)
41, 2, 3elmsubrn 31551 . . . . 5 ran 𝑆 = ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
54eleq2i 2722 . . . 4 (𝐹 ∈ ran 𝑆𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
6 eqid 2651 . . . . 5 (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) = (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
7 fvex 6239 . . . . . . 7 (mEx‘𝑇) ∈ V
81, 7eqeltri 2726 . . . . . 6 𝐸 ∈ V
98mptex 6527 . . . . 5 (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) ∈ V
106, 9elrnmpti 5408 . . . 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 1082 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝑓 ∈ ran (mRSubst‘𝑇))
13 simp3 1083 . . . . . . . . . . 11 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝑋𝐸)
14 eqid 2651 . . . . . . . . . . . 12 (mTC‘𝑇) = (mTC‘𝑇)
15 eqid 2651 . . . . . . . . . . . 12 (mREx‘𝑇) = (mREx‘𝑇)
1614, 1, 15mexval 31525 . . . . . . . . . . 11 𝐸 = ((mTC‘𝑇) × (mREx‘𝑇))
1713, 16syl6eleq 2740 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝑋 ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
18 xp2nd 7243 . . . . . . . . . 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 2651 . . . . . . . . . 10 (mVR‘𝑇) = (mVR‘𝑇)
212, 20, 15mrsubvrs 31545 . . . . . . . . 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 694 . . . . . . . 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 6229 . . . . . . . . . . . . 13 (𝑒 = 𝑋 → (1st𝑒) = (1st𝑋))
24 fveq2 6229 . . . . . . . . . . . . . 14 (𝑒 = 𝑋 → (2nd𝑒) = (2nd𝑋))
2524fveq2d 6233 . . . . . . . . . . . . 13 (𝑒 = 𝑋 → (𝑓‘(2nd𝑒)) = (𝑓‘(2nd𝑋)))
2623, 25opeq12d 4441 . . . . . . . . . . . 12 (𝑒 = 𝑋 → ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩ = ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩)
27 eqid 2651 . . . . . . . . . . . 12 (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)
28 opex 4962 . . . . . . . . . . . 12 ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩ ∈ V
2926, 27, 28fvmpt3i 6326 . . . . . . . . . . 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 6233 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋)) = (𝑉‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩))
32 xp1st 7242 . . . . . . . . . . . . 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 31540 . . . . . . . . . . . . . 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 2748 . . . . . . . . . . . . . 14 (𝑋𝐸 → (2nd𝑋) ∈ (mREx‘𝑇))
3713, 36syl 17 . . . . . . . . . . . . 13 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (2nd𝑋) ∈ (mREx‘𝑇))
3835, 37ffvelrnd 6400 . . . . . . . . . . . 12 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑓‘(2nd𝑋)) ∈ (mREx‘𝑇))
39 opelxpi 5182 . . . . . . . . . . . 12 (((1st𝑋) ∈ (mTC‘𝑇) ∧ (𝑓‘(2nd𝑋)) ∈ (mREx‘𝑇)) → ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
4033, 38, 39syl2anc 694 . . . . . . . . . . 11 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
4140, 16syl6eleqr 2741 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → ⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩ ∈ 𝐸)
42 msubvrs.v . . . . . . . . . . 11 𝑉 = (mVars‘𝑇)
4320, 1, 42mvrsval 31528 . . . . . . . . . 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 6239 . . . . . . . . . . . . 13 (1st𝑋) ∈ V
46 fvex 6239 . . . . . . . . . . . . 13 (𝑓‘(2nd𝑋)) ∈ V
4745, 46op2nd 7219 . . . . . . . . . . . 12 (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) = (𝑓‘(2nd𝑋))
4847a1i 11 . . . . . . . . . . 11 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) = (𝑓‘(2nd𝑋)))
4948rneqd 5385 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → ran (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) = ran (𝑓‘(2nd𝑋)))
5049ineq1d 3846 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (ran (2nd ‘⟨(1st𝑋), (𝑓‘(2nd𝑋))⟩) ∩ (mVR‘𝑇)) = (ran (𝑓‘(2nd𝑋)) ∩ (mVR‘𝑇)))
5131, 44, 503eqtrd 2689 . . . . . . . 8 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋)) = (ran (𝑓‘(2nd𝑋)) ∩ (mVR‘𝑇)))
5220, 1, 42mvrsval 31528 . . . . . . . . . . 11 (𝑋𝐸 → (𝑉𝑋) = (ran (2nd𝑋) ∩ (mVR‘𝑇)))
5313, 52syl 17 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉𝑋) = (ran (2nd𝑋) ∩ (mVR‘𝑇)))
5453iuneq1d 4577 . . . . . . . . 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 31581 . . . . . . . . . . . . . . . 16 (𝑇 ∈ mFS → 𝐻:(mVR‘𝑇)⟶𝐸)
57563ad2ant1 1102 . . . . . . . . . . . . . . 15 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → 𝐻:(mVR‘𝑇)⟶𝐸)
58 inss2 3867 . . . . . . . . . . . . . . . 16 (ran (2nd𝑋) ∩ (mVR‘𝑇)) ⊆ (mVR‘𝑇)
5958sseli 3632 . . . . . . . . . . . . . . 15 (𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇)) → 𝑥 ∈ (mVR‘𝑇))
60 ffvelrn 6397 . . . . . . . . . . . . . . 15 ((𝐻:(mVR‘𝑇)⟶𝐸𝑥 ∈ (mVR‘𝑇)) → (𝐻𝑥) ∈ 𝐸)
6157, 59, 60syl2an 493 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (𝐻𝑥) ∈ 𝐸)
62 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑒 = (𝐻𝑥) → (1st𝑒) = (1st ‘(𝐻𝑥)))
63 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑒 = (𝐻𝑥) → (2nd𝑒) = (2nd ‘(𝐻𝑥)))
6463fveq2d 6233 . . . . . . . . . . . . . . . 16 (𝑒 = (𝐻𝑥) → (𝑓‘(2nd𝑒)) = (𝑓‘(2nd ‘(𝐻𝑥))))
6562, 64opeq12d 4441 . . . . . . . . . . . . . . 15 (𝑒 = (𝐻𝑥) → ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩ = ⟨(1st ‘(𝐻𝑥)), (𝑓‘(2nd ‘(𝐻𝑥)))⟩)
6665, 27, 28fvmpt3i 6326 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → 𝑥 ∈ (mVR‘𝑇))
69 eqid 2651 . . . . . . . . . . . . . . . . 17 (mType‘𝑇) = (mType‘𝑇)
7020, 69, 55mvhval 31557 . . . . . . . . . . . . . . . 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 6239 . . . . . . . . . . . . . . . 16 ((mType‘𝑇)‘𝑥) ∈ V
73 s1cli 13421 . . . . . . . . . . . . . . . . 17 ⟨“𝑥”⟩ ∈ Word V
7473elexi 3244 . . . . . . . . . . . . . . . 16 ⟨“𝑥”⟩ ∈ V
7572, 74op1std 7220 . . . . . . . . . . . . . . 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 7221 . . . . . . . . . . . . . . . 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 6233 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (𝑓‘(2nd ‘(𝐻𝑥))) = (𝑓‘⟨“𝑥”⟩))
8076, 79opeq12d 4441 . . . . . . . . . . . . 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 2685 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥)) = ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩)
8281fveq2d 6233 . . . . . . . . . . 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 1084 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → 𝑇 ∈ mFS)
8420, 14, 69mtyf2 31574 . . . . . . . . . . . . . . . 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 6400 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ((mType‘𝑇)‘𝑥) ∈ (mTC‘𝑇))
8735adantr 480 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → 𝑓:(mREx‘𝑇)⟶(mREx‘𝑇))
88 elun2 3814 . . . . . . . . . . . . . . . . . 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 13419 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ⟨“𝑥”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
91 eqid 2651 . . . . . . . . . . . . . . . . . 18 (mCN‘𝑇) = (mCN‘𝑇)
9291, 20, 15mrexval 31524 . . . . . . . . . . . . . . . . 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 2733 . . . . . . . . . . . . . . 15 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ⟨“𝑥”⟩ ∈ (mREx‘𝑇))
9587, 94ffvelrnd 6400 . . . . . . . . . . . . . 14 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → (𝑓‘⟨“𝑥”⟩) ∈ (mREx‘𝑇))
96 opelxpi 5182 . . . . . . . . . . . . . 14 ((((mType‘𝑇)‘𝑥) ∈ (mTC‘𝑇) ∧ (𝑓‘⟨“𝑥”⟩) ∈ (mREx‘𝑇)) → ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
9786, 95, 96syl2anc 694 . . . . . . . . . . . . 13 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
9897, 16syl6eleqr 2741 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩ ∈ 𝐸)
9920, 1, 42mvrsval 31528 . . . . . . . . . . . 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 6239 . . . . . . . . . . . . . . 15 (𝑓‘⟨“𝑥”⟩) ∈ V
10272, 101op2nd 7219 . . . . . . . . . . . . . 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 5385 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) ∧ 𝑥 ∈ (ran (2nd𝑋) ∩ (mVR‘𝑇))) → ran (2nd ‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) = ran (𝑓‘⟨“𝑥”⟩))
105104ineq1d 3846 . . . . . . . . . . 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 2689 . . . . . . . . . 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 4574 . . . . . . . . 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 2685 . . . . . . . 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 2695 . . . . . . 7 ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋𝐸) → (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))))
110 fveq1 6228 . . . . . . . . 9 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝐹𝑋) = ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋))
111110fveq2d 6233 . . . . . . . 8 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑉‘(𝐹𝑋)) = (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘𝑋)))
112 fveq1 6228 . . . . . . . . . 10 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝐹‘(𝐻𝑥)) = ((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥)))
113112fveq2d 6233 . . . . . . . . 9 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → (𝑉‘(𝐹‘(𝐻𝑥))) = (𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))))
114113iuneq2d 4579 . . . . . . . 8 (𝐹 = (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) → 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))) = 𝑥 ∈ (𝑉𝑋)(𝑉‘((𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)‘(𝐻𝑥))))
115111, 114eqeq12d 2666 . . . . . . 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 1286 . . . . 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 3060 . . 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 1275 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 1054   = wceq 1523  wcel 2030  wrex 2942  Vcvv 3231  cun 3605  cin 3606  cop 4216   ciun 4552  cmpt 4762   × cxp 5141  ran crn 5144  wf 5922  cfv 5926  1st c1st 7208  2nd c2nd 7209  Word cword 13323  ⟨“cs1 13326  mCNcmcn 31483  mVRcmvar 31484  mTypecmty 31485  mTCcmtc 31487  mRExcmrex 31489  mExcmex 31490  mVarscmvrs 31492  mRSubstcmrsub 31493  mSubstcmsub 31494  mVHcmvh 31495  mFScmfs 31499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-word 13331  df-lsw 13332  df-concat 13333  df-s1 13334  df-substr 13335  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-0g 16149  df-gsum 16150  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-frmd 17433  df-mrex 31509  df-mex 31510  df-mvrs 31512  df-mrsub 31513  df-msub 31514  df-mvh 31515  df-mfs 31519
This theorem is referenced by:  mclsppslem  31606
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