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Theorem mrsubvrs 31758
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
mrsubco.s 𝑆 = (mRSubst‘𝑇)
mrsubvrs.v 𝑉 = (mVR‘𝑇)
mrsubvrs.r 𝑅 = (mREx‘𝑇)
Assertion
Ref Expression
mrsubvrs ((𝐹 ∈ ran 𝑆𝑋𝑅) → (ran (𝐹𝑋) ∩ 𝑉) = 𝑥 ∈ (ran 𝑋𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑆   𝑥,𝑇   𝑥,𝑉   𝑥,𝑋
Allowed substitution hint:   𝑅(𝑥)

Proof of Theorem mrsubvrs
Dummy variables 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0i 4069 . . . . . 6 (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
2 mrsubco.s . . . . . . . . 9 𝑆 = (mRSubst‘𝑇)
3 fvprc 6327 . . . . . . . . 9 𝑇 ∈ V → (mRSubst‘𝑇) = ∅)
42, 3syl5eq 2817 . . . . . . . 8 𝑇 ∈ V → 𝑆 = ∅)
54rneqd 5492 . . . . . . 7 𝑇 ∈ V → ran 𝑆 = ran ∅)
6 rn0 5516 . . . . . . 7 ran ∅ = ∅
75, 6syl6eq 2821 . . . . . 6 𝑇 ∈ V → ran 𝑆 = ∅)
81, 7nsyl2 144 . . . . 5 (𝐹 ∈ ran 𝑆𝑇 ∈ V)
9 eqid 2771 . . . . . 6 (mCN‘𝑇) = (mCN‘𝑇)
10 mrsubvrs.v . . . . . 6 𝑉 = (mVR‘𝑇)
11 mrsubvrs.r . . . . . 6 𝑅 = (mREx‘𝑇)
129, 10, 11mrexval 31737 . . . . 5 (𝑇 ∈ V → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
138, 12syl 17 . . . 4 (𝐹 ∈ ran 𝑆𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
1413eleq2d 2836 . . 3 (𝐹 ∈ ran 𝑆 → (𝑋𝑅𝑋 ∈ Word ((mCN‘𝑇) ∪ 𝑉)))
15 fveq2 6333 . . . . . . . . 9 (𝑣 = ∅ → (𝐹𝑣) = (𝐹‘∅))
1615rneqd 5492 . . . . . . . 8 (𝑣 = ∅ → ran (𝐹𝑣) = ran (𝐹‘∅))
1716ineq1d 3965 . . . . . . 7 (𝑣 = ∅ → (ran (𝐹𝑣) ∩ 𝑉) = (ran (𝐹‘∅) ∩ 𝑉))
18 rneq 5490 . . . . . . . . . . . 12 (𝑣 = ∅ → ran 𝑣 = ran ∅)
1918, 6syl6eq 2821 . . . . . . . . . . 11 (𝑣 = ∅ → ran 𝑣 = ∅)
2019ineq1d 3965 . . . . . . . . . 10 (𝑣 = ∅ → (ran 𝑣𝑉) = (∅ ∩ 𝑉))
21 0in 4114 . . . . . . . . . 10 (∅ ∩ 𝑉) = ∅
2220, 21syl6eq 2821 . . . . . . . . 9 (𝑣 = ∅ → (ran 𝑣𝑉) = ∅)
2322iuneq1d 4680 . . . . . . . 8 (𝑣 = ∅ → 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ ∅ (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
24 0iun 4712 . . . . . . . 8 𝑥 ∈ ∅ (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∅
2523, 24syl6eq 2821 . . . . . . 7 (𝑣 = ∅ → 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∅)
2617, 25eqeq12d 2786 . . . . . 6 (𝑣 = ∅ → ((ran (𝐹𝑣) ∩ 𝑉) = 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ↔ (ran (𝐹‘∅) ∩ 𝑉) = ∅))
2726imbi2d 329 . . . . 5 (𝑣 = ∅ → ((𝐹 ∈ ran 𝑆 → (ran (𝐹𝑣) ∩ 𝑉) = 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)) ↔ (𝐹 ∈ ran 𝑆 → (ran (𝐹‘∅) ∩ 𝑉) = ∅)))
28 fveq2 6333 . . . . . . . . 9 (𝑣 = 𝑦 → (𝐹𝑣) = (𝐹𝑦))
2928rneqd 5492 . . . . . . . 8 (𝑣 = 𝑦 → ran (𝐹𝑣) = ran (𝐹𝑦))
3029ineq1d 3965 . . . . . . 7 (𝑣 = 𝑦 → (ran (𝐹𝑣) ∩ 𝑉) = (ran (𝐹𝑦) ∩ 𝑉))
31 rneq 5490 . . . . . . . . 9 (𝑣 = 𝑦 → ran 𝑣 = ran 𝑦)
3231ineq1d 3965 . . . . . . . 8 (𝑣 = 𝑦 → (ran 𝑣𝑉) = (ran 𝑦𝑉))
3332iuneq1d 4680 . . . . . . 7 (𝑣 = 𝑦 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
3430, 33eqeq12d 2786 . . . . . 6 (𝑣 = 𝑦 → ((ran (𝐹𝑣) ∩ 𝑉) = 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ↔ (ran (𝐹𝑦) ∩ 𝑉) = 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
3534imbi2d 329 . . . . 5 (𝑣 = 𝑦 → ((𝐹 ∈ ran 𝑆 → (ran (𝐹𝑣) ∩ 𝑉) = 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)) ↔ (𝐹 ∈ ran 𝑆 → (ran (𝐹𝑦) ∩ 𝑉) = 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))))
36 fveq2 6333 . . . . . . . . 9 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → (𝐹𝑣) = (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)))
3736rneqd 5492 . . . . . . . 8 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → ran (𝐹𝑣) = ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)))
3837ineq1d 3965 . . . . . . 7 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → (ran (𝐹𝑣) ∩ 𝑉) = (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉))
39 rneq 5490 . . . . . . . . 9 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → ran 𝑣 = ran (𝑦 ++ ⟨“𝑧”⟩))
4039ineq1d 3965 . . . . . . . 8 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → (ran 𝑣𝑉) = (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉))
4140iuneq1d 4680 . . . . . . 7 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
4238, 41eqeq12d 2786 . . . . . 6 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → ((ran (𝐹𝑣) ∩ 𝑉) = 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ↔ (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉) = 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
4342imbi2d 329 . . . . 5 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → ((𝐹 ∈ ran 𝑆 → (ran (𝐹𝑣) ∩ 𝑉) = 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)) ↔ (𝐹 ∈ ran 𝑆 → (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉) = 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))))
44 fveq2 6333 . . . . . . . . 9 (𝑣 = 𝑋 → (𝐹𝑣) = (𝐹𝑋))
4544rneqd 5492 . . . . . . . 8 (𝑣 = 𝑋 → ran (𝐹𝑣) = ran (𝐹𝑋))
4645ineq1d 3965 . . . . . . 7 (𝑣 = 𝑋 → (ran (𝐹𝑣) ∩ 𝑉) = (ran (𝐹𝑋) ∩ 𝑉))
47 rneq 5490 . . . . . . . . 9 (𝑣 = 𝑋 → ran 𝑣 = ran 𝑋)
4847ineq1d 3965 . . . . . . . 8 (𝑣 = 𝑋 → (ran 𝑣𝑉) = (ran 𝑋𝑉))
4948iuneq1d 4680 . . . . . . 7 (𝑣 = 𝑋 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ (ran 𝑋𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
5046, 49eqeq12d 2786 . . . . . 6 (𝑣 = 𝑋 → ((ran (𝐹𝑣) ∩ 𝑉) = 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ↔ (ran (𝐹𝑋) ∩ 𝑉) = 𝑥 ∈ (ran 𝑋𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
5150imbi2d 329 . . . . 5 (𝑣 = 𝑋 → ((𝐹 ∈ ran 𝑆 → (ran (𝐹𝑣) ∩ 𝑉) = 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)) ↔ (𝐹 ∈ ran 𝑆 → (ran (𝐹𝑋) ∩ 𝑉) = 𝑥 ∈ (ran 𝑋𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))))
522mrsub0 31752 . . . . . . . . 9 (𝐹 ∈ ran 𝑆 → (𝐹‘∅) = ∅)
5352rneqd 5492 . . . . . . . 8 (𝐹 ∈ ran 𝑆 → ran (𝐹‘∅) = ran ∅)
5453, 6syl6eq 2821 . . . . . . 7 (𝐹 ∈ ran 𝑆 → ran (𝐹‘∅) = ∅)
5554ineq1d 3965 . . . . . 6 (𝐹 ∈ ran 𝑆 → (ran (𝐹‘∅) ∩ 𝑉) = (∅ ∩ 𝑉))
5655, 21syl6eq 2821 . . . . 5 (𝐹 ∈ ran 𝑆 → (ran (𝐹‘∅) ∩ 𝑉) = ∅)
57 uneq1 3912 . . . . . . . 8 ((ran (𝐹𝑦) ∩ 𝑉) = 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) → ((ran (𝐹𝑦) ∩ 𝑉) ∪ (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉)) = ( 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ∪ (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉)))
58 simpl 468 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝐹 ∈ ran 𝑆)
59 simprl 748 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉))
6013adantr 466 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
6159, 60eleqtrrd 2853 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑦𝑅)
62 simprr 750 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))
6362s1cld 13584 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ⟨“𝑧”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉))
6463, 60eleqtrrd 2853 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ⟨“𝑧”⟩ ∈ 𝑅)
652, 11mrsubccat 31754 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆𝑦𝑅 ∧ ⟨“𝑧”⟩ ∈ 𝑅) → (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = ((𝐹𝑦) ++ (𝐹‘⟨“𝑧”⟩)))
6658, 61, 64, 65syl3anc 1476 . . . . . . . . . . . . 13 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = ((𝐹𝑦) ++ (𝐹‘⟨“𝑧”⟩)))
6766rneqd 5492 . . . . . . . . . . . 12 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = ran ((𝐹𝑦) ++ (𝐹‘⟨“𝑧”⟩)))
682, 11mrsubf 31753 . . . . . . . . . . . . . . . 16 (𝐹 ∈ ran 𝑆𝐹:𝑅𝑅)
6968adantr 466 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝐹:𝑅𝑅)
7069, 61ffvelrnd 6504 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹𝑦) ∈ 𝑅)
7170, 60eleqtrd 2852 . . . . . . . . . . . . 13 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹𝑦) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
7269, 64ffvelrnd 6504 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘⟨“𝑧”⟩) ∈ 𝑅)
7372, 60eleqtrd 2852 . . . . . . . . . . . . 13 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘⟨“𝑧”⟩) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
74 ccatrn 13572 . . . . . . . . . . . . 13 (((𝐹𝑦) ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ (𝐹‘⟨“𝑧”⟩) ∈ Word ((mCN‘𝑇) ∪ 𝑉)) → ran ((𝐹𝑦) ++ (𝐹‘⟨“𝑧”⟩)) = (ran (𝐹𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)))
7571, 73, 74syl2anc 567 . . . . . . . . . . . 12 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran ((𝐹𝑦) ++ (𝐹‘⟨“𝑧”⟩)) = (ran (𝐹𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)))
7667, 75eqtrd 2805 . . . . . . . . . . 11 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = (ran (𝐹𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)))
7776ineq1d 3965 . . . . . . . . . 10 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉) = ((ran (𝐹𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)) ∩ 𝑉))
78 indir 4025 . . . . . . . . . 10 ((ran (𝐹𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)) ∩ 𝑉) = ((ran (𝐹𝑦) ∩ 𝑉) ∪ (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
7977, 78syl6eq 2821 . . . . . . . . 9 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉) = ((ran (𝐹𝑦) ∩ 𝑉) ∪ (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉)))
80 ccatrn 13572 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ ⟨“𝑧”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉)) → ran (𝑦 ++ ⟨“𝑧”⟩) = (ran 𝑦 ∪ ran ⟨“𝑧”⟩))
8159, 63, 80syl2anc 567 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝑦 ++ ⟨“𝑧”⟩) = (ran 𝑦 ∪ ran ⟨“𝑧”⟩))
82 s1rn 13580 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉) → ran ⟨“𝑧”⟩ = {𝑧})
8382ad2antll 702 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran ⟨“𝑧”⟩ = {𝑧})
8483uneq2d 3919 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran 𝑦 ∪ ran ⟨“𝑧”⟩) = (ran 𝑦 ∪ {𝑧}))
8581, 84eqtrd 2805 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝑦 ++ ⟨“𝑧”⟩) = (ran 𝑦 ∪ {𝑧}))
8685ineq1d 3965 . . . . . . . . . . . . 13 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉) = ((ran 𝑦 ∪ {𝑧}) ∩ 𝑉))
87 indir 4025 . . . . . . . . . . . . 13 ((ran 𝑦 ∪ {𝑧}) ∩ 𝑉) = ((ran 𝑦𝑉) ∪ ({𝑧} ∩ 𝑉))
8886, 87syl6eq 2821 . . . . . . . . . . . 12 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉) = ((ran 𝑦𝑉) ∪ ({𝑧} ∩ 𝑉)))
8988iuneq1d 4680 . . . . . . . . . . 11 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ ((ran 𝑦𝑉) ∪ ({𝑧} ∩ 𝑉))(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
90 iunxun 4740 . . . . . . . . . . 11 𝑥 ∈ ((ran 𝑦𝑉) ∪ ({𝑧} ∩ 𝑉))(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ( 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ∪ 𝑥 ∈ ({𝑧} ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
9189, 90syl6eq 2821 . . . . . . . . . 10 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ( 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ∪ 𝑥 ∈ ({𝑧} ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
92 simpr 471 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧𝑉) → 𝑧𝑉)
9392snssd 4476 . . . . . . . . . . . . . . 15 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧𝑉) → {𝑧} ⊆ 𝑉)
94 df-ss 3738 . . . . . . . . . . . . . . 15 ({𝑧} ⊆ 𝑉 ↔ ({𝑧} ∩ 𝑉) = {𝑧})
9593, 94sylib 208 . . . . . . . . . . . . . 14 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧𝑉) → ({𝑧} ∩ 𝑉) = {𝑧})
9695iuneq1d 4680 . . . . . . . . . . . . 13 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧𝑉) → 𝑥 ∈ ({𝑧} ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ {𝑧} (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
97 vex 3354 . . . . . . . . . . . . . 14 𝑧 ∈ V
98 s1eq 13581 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ⟨“𝑥”⟩ = ⟨“𝑧”⟩)
9998fveq2d 6337 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (𝐹‘⟨“𝑥”⟩) = (𝐹‘⟨“𝑧”⟩))
10099rneqd 5492 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → ran (𝐹‘⟨“𝑥”⟩) = ran (𝐹‘⟨“𝑧”⟩))
101100ineq1d 3965 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
10297, 101iunxsn 4738 . . . . . . . . . . . . 13 𝑥 ∈ {𝑧} (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉)
10396, 102syl6eq 2821 . . . . . . . . . . . 12 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧𝑉) → 𝑥 ∈ ({𝑧} ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
104 incom 3957 . . . . . . . . . . . . . . 15 ({𝑧} ∩ 𝑉) = (𝑉 ∩ {𝑧})
105 simpr 471 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ¬ 𝑧𝑉)
106 disjsn 4384 . . . . . . . . . . . . . . . 16 ((𝑉 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑉)
107105, 106sylibr 224 . . . . . . . . . . . . . . 15 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → (𝑉 ∩ {𝑧}) = ∅)
108104, 107syl5eq 2817 . . . . . . . . . . . . . 14 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ({𝑧} ∩ 𝑉) = ∅)
109108iuneq1d 4680 . . . . . . . . . . . . 13 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → 𝑥 ∈ ({𝑧} ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ ∅ (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
11058adantr 466 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → 𝐹 ∈ ran 𝑆)
111 eldif 3734 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉) ↔ (𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉) ∧ ¬ 𝑧𝑉))
112111biimpri 218 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉) ∧ ¬ 𝑧𝑉) → 𝑧 ∈ (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉))
11362, 112sylan 563 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → 𝑧 ∈ (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉))
114 difun2 4191 . . . . . . . . . . . . . . . . . . 19 (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉) = ((mCN‘𝑇) ∖ 𝑉)
115113, 114syl6eleq 2860 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → 𝑧 ∈ ((mCN‘𝑇) ∖ 𝑉))
1162, 11, 10, 9mrsubcn 31755 . . . . . . . . . . . . . . . . . 18 ((𝐹 ∈ ran 𝑆𝑧 ∈ ((mCN‘𝑇) ∖ 𝑉)) → (𝐹‘⟨“𝑧”⟩) = ⟨“𝑧”⟩)
117110, 115, 116syl2anc 567 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → (𝐹‘⟨“𝑧”⟩) = ⟨“𝑧”⟩)
118117rneqd 5492 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ran (𝐹‘⟨“𝑧”⟩) = ran ⟨“𝑧”⟩)
11983adantr 466 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ran ⟨“𝑧”⟩ = {𝑧})
120118, 119eqtrd 2805 . . . . . . . . . . . . . . 15 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ran (𝐹‘⟨“𝑧”⟩) = {𝑧})
121120ineq1d 3965 . . . . . . . . . . . . . 14 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉) = ({𝑧} ∩ 𝑉))
122121, 108eqtrd 2805 . . . . . . . . . . . . 13 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉) = ∅)
12324, 109, 1223eqtr4a 2831 . . . . . . . . . . . 12 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → 𝑥 ∈ ({𝑧} ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
124103, 123pm2.61dan 807 . . . . . . . . . . 11 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑥 ∈ ({𝑧} ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
125124uneq2d 3919 . . . . . . . . . 10 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ( 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ∪ 𝑥 ∈ ({𝑧} ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)) = ( 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ∪ (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉)))
12691, 125eqtrd 2805 . . . . . . . . 9 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ( 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ∪ (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉)))
12779, 126eqeq12d 2786 . . . . . . . 8 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ((ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉) = 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ↔ ((ran (𝐹𝑦) ∩ 𝑉) ∪ (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉)) = ( 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ∪ (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))))
12857, 127syl5ibr 236 . . . . . . 7 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ((ran (𝐹𝑦) ∩ 𝑉) = 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) → (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉) = 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
129128expcom 398 . . . . . 6 ((𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉)) → (𝐹 ∈ ran 𝑆 → ((ran (𝐹𝑦) ∩ 𝑉) = 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) → (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉) = 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))))
130129a2d 29 . . . . 5 ((𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉)) → ((𝐹 ∈ ran 𝑆 → (ran (𝐹𝑦) ∩ 𝑉) = 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)) → (𝐹 ∈ ran 𝑆 → (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉) = 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))))
13127, 35, 43, 51, 56, 130wrdind 13686 . . . 4 (𝑋 ∈ Word ((mCN‘𝑇) ∪ 𝑉) → (𝐹 ∈ ran 𝑆 → (ran (𝐹𝑋) ∩ 𝑉) = 𝑥 ∈ (ran 𝑋𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
132131com12 32 . . 3 (𝐹 ∈ ran 𝑆 → (𝑋 ∈ Word ((mCN‘𝑇) ∪ 𝑉) → (ran (𝐹𝑋) ∩ 𝑉) = 𝑥 ∈ (ran 𝑋𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
13314, 132sylbid 230 . 2 (𝐹 ∈ ran 𝑆 → (𝑋𝑅 → (ran (𝐹𝑋) ∩ 𝑉) = 𝑥 ∈ (ran 𝑋𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
134133imp 393 1 ((𝐹 ∈ ran 𝑆𝑋𝑅) → (ran (𝐹𝑋) ∩ 𝑉) = 𝑥 ∈ (ran 𝑋𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cdif 3721  cun 3722  cin 3723  wss 3724  c0 4064  {csn 4317   ciun 4655  ran crn 5251  wf 6028  cfv 6032  (class class class)co 6794  Word cword 13488   ++ cconcat 13490  ⟨“cs1 13491  mCNcmcn 31696  mVRcmvar 31697  mRExcmrex 31702  mRSubstcmrsub 31706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097  ax-cnex 10195  ax-resscn 10196  ax-1cn 10197  ax-icn 10198  ax-addcl 10199  ax-addrcl 10200  ax-mulcl 10201  ax-mulrcl 10202  ax-mulcom 10203  ax-addass 10204  ax-mulass 10205  ax-distr 10206  ax-i2m1 10207  ax-1ne0 10208  ax-1rid 10209  ax-rnegex 10210  ax-rrecex 10211  ax-cnre 10212  ax-pre-lttri 10213  ax-pre-lttrn 10214  ax-pre-ltadd 10215  ax-pre-mulgt0 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-riota 6755  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-om 7214  df-1st 7316  df-2nd 7317  df-wrecs 7560  df-recs 7622  df-rdg 7660  df-1o 7714  df-oadd 7718  df-er 7897  df-map 8012  df-pm 8013  df-en 8111  df-dom 8112  df-sdom 8113  df-fin 8114  df-card 8966  df-pnf 10279  df-mnf 10280  df-xr 10281  df-ltxr 10282  df-le 10283  df-sub 10471  df-neg 10472  df-nn 11224  df-2 11282  df-n0 11496  df-xnn0 11567  df-z 11581  df-uz 11890  df-fz 12535  df-fzo 12675  df-seq 13010  df-hash 13323  df-word 13496  df-lsw 13497  df-concat 13498  df-s1 13499  df-substr 13500  df-struct 16067  df-ndx 16068  df-slot 16069  df-base 16071  df-sets 16072  df-ress 16073  df-plusg 16163  df-0g 16311  df-gsum 16312  df-mgm 17451  df-sgrp 17493  df-mnd 17504  df-submnd 17545  df-frmd 17595  df-mrex 31722  df-mrsub 31726
This theorem is referenced by:  msubvrs  31796
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