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Theorem mrsubvrs 34116
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 4293 . . . . . 6 (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
2 mrsubco.s . . . . . . 7 𝑆 = (mRSubst‘𝑇)
32rnfvprc 6836 . . . . . 6 𝑇 ∈ V → ran 𝑆 = ∅)
41, 3nsyl2 141 . . . . 5 (𝐹 ∈ ran 𝑆𝑇 ∈ V)
5 eqid 2736 . . . . . 6 (mCN‘𝑇) = (mCN‘𝑇)
6 mrsubvrs.v . . . . . 6 𝑉 = (mVR‘𝑇)
7 mrsubvrs.r . . . . . 6 𝑅 = (mREx‘𝑇)
85, 6, 7mrexval 34095 . . . . 5 (𝑇 ∈ V → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
94, 8syl 17 . . . 4 (𝐹 ∈ ran 𝑆𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
109eleq2d 2823 . . 3 (𝐹 ∈ ran 𝑆 → (𝑋𝑅𝑋 ∈ Word ((mCN‘𝑇) ∪ 𝑉)))
11 fveq2 6842 . . . . . . . . 9 (𝑣 = ∅ → (𝐹𝑣) = (𝐹‘∅))
1211rneqd 5893 . . . . . . . 8 (𝑣 = ∅ → ran (𝐹𝑣) = ran (𝐹‘∅))
1312ineq1d 4171 . . . . . . 7 (𝑣 = ∅ → (ran (𝐹𝑣) ∩ 𝑉) = (ran (𝐹‘∅) ∩ 𝑉))
14 rneq 5891 . . . . . . . . . . . 12 (𝑣 = ∅ → ran 𝑣 = ran ∅)
15 rn0 5881 . . . . . . . . . . . 12 ran ∅ = ∅
1614, 15eqtrdi 2792 . . . . . . . . . . 11 (𝑣 = ∅ → ran 𝑣 = ∅)
1716ineq1d 4171 . . . . . . . . . 10 (𝑣 = ∅ → (ran 𝑣𝑉) = (∅ ∩ 𝑉))
18 0in 4353 . . . . . . . . . 10 (∅ ∩ 𝑉) = ∅
1917, 18eqtrdi 2792 . . . . . . . . 9 (𝑣 = ∅ → (ran 𝑣𝑉) = ∅)
2019iuneq1d 4981 . . . . . . . 8 (𝑣 = ∅ → 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ ∅ (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
21 0iun 5023 . . . . . . . 8 𝑥 ∈ ∅ (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∅
2220, 21eqtrdi 2792 . . . . . . 7 (𝑣 = ∅ → 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∅)
2313, 22eqeq12d 2752 . . . . . 6 (𝑣 = ∅ → ((ran (𝐹𝑣) ∩ 𝑉) = 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ↔ (ran (𝐹‘∅) ∩ 𝑉) = ∅))
2423imbi2d 340 . . . . 5 (𝑣 = ∅ → ((𝐹 ∈ ran 𝑆 → (ran (𝐹𝑣) ∩ 𝑉) = 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)) ↔ (𝐹 ∈ ran 𝑆 → (ran (𝐹‘∅) ∩ 𝑉) = ∅)))
25 fveq2 6842 . . . . . . . . 9 (𝑣 = 𝑦 → (𝐹𝑣) = (𝐹𝑦))
2625rneqd 5893 . . . . . . . 8 (𝑣 = 𝑦 → ran (𝐹𝑣) = ran (𝐹𝑦))
2726ineq1d 4171 . . . . . . 7 (𝑣 = 𝑦 → (ran (𝐹𝑣) ∩ 𝑉) = (ran (𝐹𝑦) ∩ 𝑉))
28 rneq 5891 . . . . . . . . 9 (𝑣 = 𝑦 → ran 𝑣 = ran 𝑦)
2928ineq1d 4171 . . . . . . . 8 (𝑣 = 𝑦 → (ran 𝑣𝑉) = (ran 𝑦𝑉))
3029iuneq1d 4981 . . . . . . 7 (𝑣 = 𝑦 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
3127, 30eqeq12d 2752 . . . . . 6 (𝑣 = 𝑦 → ((ran (𝐹𝑣) ∩ 𝑉) = 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ↔ (ran (𝐹𝑦) ∩ 𝑉) = 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
3231imbi2d 340 . . . . 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))))
33 fveq2 6842 . . . . . . . . 9 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → (𝐹𝑣) = (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)))
3433rneqd 5893 . . . . . . . 8 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → ran (𝐹𝑣) = ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)))
3534ineq1d 4171 . . . . . . 7 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → (ran (𝐹𝑣) ∩ 𝑉) = (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉))
36 rneq 5891 . . . . . . . . 9 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → ran 𝑣 = ran (𝑦 ++ ⟨“𝑧”⟩))
3736ineq1d 4171 . . . . . . . 8 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → (ran 𝑣𝑉) = (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉))
3837iuneq1d 4981 . . . . . . 7 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
3935, 38eqeq12d 2752 . . . . . 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)))
4039imbi2d 340 . . . . 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))))
41 fveq2 6842 . . . . . . . . 9 (𝑣 = 𝑋 → (𝐹𝑣) = (𝐹𝑋))
4241rneqd 5893 . . . . . . . 8 (𝑣 = 𝑋 → ran (𝐹𝑣) = ran (𝐹𝑋))
4342ineq1d 4171 . . . . . . 7 (𝑣 = 𝑋 → (ran (𝐹𝑣) ∩ 𝑉) = (ran (𝐹𝑋) ∩ 𝑉))
44 rneq 5891 . . . . . . . . 9 (𝑣 = 𝑋 → ran 𝑣 = ran 𝑋)
4544ineq1d 4171 . . . . . . . 8 (𝑣 = 𝑋 → (ran 𝑣𝑉) = (ran 𝑋𝑉))
4645iuneq1d 4981 . . . . . . 7 (𝑣 = 𝑋 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ (ran 𝑋𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
4743, 46eqeq12d 2752 . . . . . 6 (𝑣 = 𝑋 → ((ran (𝐹𝑣) ∩ 𝑉) = 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ↔ (ran (𝐹𝑋) ∩ 𝑉) = 𝑥 ∈ (ran 𝑋𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
4847imbi2d 340 . . . . 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))))
492mrsub0 34110 . . . . . . . . 9 (𝐹 ∈ ran 𝑆 → (𝐹‘∅) = ∅)
5049rneqd 5893 . . . . . . . 8 (𝐹 ∈ ran 𝑆 → ran (𝐹‘∅) = ran ∅)
5150, 15eqtrdi 2792 . . . . . . 7 (𝐹 ∈ ran 𝑆 → ran (𝐹‘∅) = ∅)
5251ineq1d 4171 . . . . . 6 (𝐹 ∈ ran 𝑆 → (ran (𝐹‘∅) ∩ 𝑉) = (∅ ∩ 𝑉))
5352, 18eqtrdi 2792 . . . . 5 (𝐹 ∈ ran 𝑆 → (ran (𝐹‘∅) ∩ 𝑉) = ∅)
54 uneq1 4116 . . . . . . . 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)))
55 simpl 483 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝐹 ∈ ran 𝑆)
56 simprl 769 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉))
579adantr 481 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
5856, 57eleqtrrd 2841 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑦𝑅)
59 simprr 771 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))
6059s1cld 14491 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ⟨“𝑧”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉))
6160, 57eleqtrrd 2841 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ⟨“𝑧”⟩ ∈ 𝑅)
622, 7mrsubccat 34112 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆𝑦𝑅 ∧ ⟨“𝑧”⟩ ∈ 𝑅) → (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = ((𝐹𝑦) ++ (𝐹‘⟨“𝑧”⟩)))
6355, 58, 61, 62syl3anc 1371 . . . . . . . . . . . . 13 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = ((𝐹𝑦) ++ (𝐹‘⟨“𝑧”⟩)))
6463rneqd 5893 . . . . . . . . . . . 12 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = ran ((𝐹𝑦) ++ (𝐹‘⟨“𝑧”⟩)))
652, 7mrsubf 34111 . . . . . . . . . . . . . . . 16 (𝐹 ∈ ran 𝑆𝐹:𝑅𝑅)
6665adantr 481 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝐹:𝑅𝑅)
6766, 58ffvelcdmd 7036 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹𝑦) ∈ 𝑅)
6867, 57eleqtrd 2840 . . . . . . . . . . . . 13 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹𝑦) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
6966, 61ffvelcdmd 7036 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘⟨“𝑧”⟩) ∈ 𝑅)
7069, 57eleqtrd 2840 . . . . . . . . . . . . 13 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘⟨“𝑧”⟩) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
71 ccatrn 14477 . . . . . . . . . . . . 13 (((𝐹𝑦) ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ (𝐹‘⟨“𝑧”⟩) ∈ Word ((mCN‘𝑇) ∪ 𝑉)) → ran ((𝐹𝑦) ++ (𝐹‘⟨“𝑧”⟩)) = (ran (𝐹𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)))
7268, 70, 71syl2anc 584 . . . . . . . . . . . 12 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran ((𝐹𝑦) ++ (𝐹‘⟨“𝑧”⟩)) = (ran (𝐹𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)))
7364, 72eqtrd 2776 . . . . . . . . . . 11 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = (ran (𝐹𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)))
7473ineq1d 4171 . . . . . . . . . 10 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉) = ((ran (𝐹𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)) ∩ 𝑉))
75 indir 4235 . . . . . . . . . 10 ((ran (𝐹𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)) ∩ 𝑉) = ((ran (𝐹𝑦) ∩ 𝑉) ∪ (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
7674, 75eqtrdi 2792 . . . . . . . . 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)))
77 ccatrn 14477 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ ⟨“𝑧”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉)) → ran (𝑦 ++ ⟨“𝑧”⟩) = (ran 𝑦 ∪ ran ⟨“𝑧”⟩))
7856, 60, 77syl2anc 584 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝑦 ++ ⟨“𝑧”⟩) = (ran 𝑦 ∪ ran ⟨“𝑧”⟩))
79 s1rn 14487 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉) → ran ⟨“𝑧”⟩ = {𝑧})
8079ad2antll 727 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran ⟨“𝑧”⟩ = {𝑧})
8180uneq2d 4123 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran 𝑦 ∪ ran ⟨“𝑧”⟩) = (ran 𝑦 ∪ {𝑧}))
8278, 81eqtrd 2776 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝑦 ++ ⟨“𝑧”⟩) = (ran 𝑦 ∪ {𝑧}))
8382ineq1d 4171 . . . . . . . . . . . . 13 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉) = ((ran 𝑦 ∪ {𝑧}) ∩ 𝑉))
84 indir 4235 . . . . . . . . . . . . 13 ((ran 𝑦 ∪ {𝑧}) ∩ 𝑉) = ((ran 𝑦𝑉) ∪ ({𝑧} ∩ 𝑉))
8583, 84eqtrdi 2792 . . . . . . . . . . . 12 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉) = ((ran 𝑦𝑉) ∪ ({𝑧} ∩ 𝑉)))
8685iuneq1d 4981 . . . . . . . . . . 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))
87 iunxun 5054 . . . . . . . . . . 11 𝑥 ∈ ((ran 𝑦𝑉) ∪ ({𝑧} ∩ 𝑉))(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ( 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ∪ 𝑥 ∈ ({𝑧} ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
8886, 87eqtrdi 2792 . . . . . . . . . 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)))
89 simpr 485 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧𝑉) → 𝑧𝑉)
9089snssd 4769 . . . . . . . . . . . . . . 15 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧𝑉) → {𝑧} ⊆ 𝑉)
91 df-ss 3927 . . . . . . . . . . . . . . 15 ({𝑧} ⊆ 𝑉 ↔ ({𝑧} ∩ 𝑉) = {𝑧})
9290, 91sylib 217 . . . . . . . . . . . . . 14 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧𝑉) → ({𝑧} ∩ 𝑉) = {𝑧})
9392iuneq1d 4981 . . . . . . . . . . . . 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))
94 vex 3449 . . . . . . . . . . . . . 14 𝑧 ∈ V
95 s1eq 14488 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ⟨“𝑥”⟩ = ⟨“𝑧”⟩)
9695fveq2d 6846 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (𝐹‘⟨“𝑥”⟩) = (𝐹‘⟨“𝑧”⟩))
9796rneqd 5893 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → ran (𝐹‘⟨“𝑥”⟩) = ran (𝐹‘⟨“𝑧”⟩))
9897ineq1d 4171 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
9994, 98iunxsn 5051 . . . . . . . . . . . . 13 𝑥 ∈ {𝑧} (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉)
10093, 99eqtrdi 2792 . . . . . . . . . . . 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))
101 incom 4161 . . . . . . . . . . . . . . 15 ({𝑧} ∩ 𝑉) = (𝑉 ∩ {𝑧})
102 simpr 485 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ¬ 𝑧𝑉)
103 disjsn 4672 . . . . . . . . . . . . . . . 16 ((𝑉 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑉)
104102, 103sylibr 233 . . . . . . . . . . . . . . 15 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → (𝑉 ∩ {𝑧}) = ∅)
105101, 104eqtrid 2788 . . . . . . . . . . . . . 14 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ({𝑧} ∩ 𝑉) = ∅)
106105iuneq1d 4981 . . . . . . . . . . . . 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))
10755adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → 𝐹 ∈ ran 𝑆)
108 eldif 3920 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉) ↔ (𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉) ∧ ¬ 𝑧𝑉))
109108biimpri 227 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉) ∧ ¬ 𝑧𝑉) → 𝑧 ∈ (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉))
11059, 109sylan 580 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → 𝑧 ∈ (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉))
111 difun2 4440 . . . . . . . . . . . . . . . . . . 19 (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉) = ((mCN‘𝑇) ∖ 𝑉)
112110, 111eleqtrdi 2848 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → 𝑧 ∈ ((mCN‘𝑇) ∖ 𝑉))
1132, 7, 6, 5mrsubcn 34113 . . . . . . . . . . . . . . . . . 18 ((𝐹 ∈ ran 𝑆𝑧 ∈ ((mCN‘𝑇) ∖ 𝑉)) → (𝐹‘⟨“𝑧”⟩) = ⟨“𝑧”⟩)
114107, 112, 113syl2anc 584 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → (𝐹‘⟨“𝑧”⟩) = ⟨“𝑧”⟩)
115114rneqd 5893 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ran (𝐹‘⟨“𝑧”⟩) = ran ⟨“𝑧”⟩)
11680adantr 481 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ran ⟨“𝑧”⟩ = {𝑧})
117115, 116eqtrd 2776 . . . . . . . . . . . . . . 15 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ran (𝐹‘⟨“𝑧”⟩) = {𝑧})
118117ineq1d 4171 . . . . . . . . . . . . . 14 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉) = ({𝑧} ∩ 𝑉))
119118, 105eqtrd 2776 . . . . . . . . . . . . 13 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉) = ∅)
12021, 106, 1193eqtr4a 2802 . . . . . . . . . . . 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))
121100, 120pm2.61dan 811 . . . . . . . . . . 11 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑥 ∈ ({𝑧} ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
122121uneq2d 4123 . . . . . . . . . 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)))
12388, 122eqtrd 2776 . . . . . . . . 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)))
12476, 123eqeq12d 2752 . . . . . . . 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))))
12554, 124syl5ibr 245 . . . . . . 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)))
126125expcom 414 . . . . . 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))))
127126a2d 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))))
12824, 32, 40, 48, 53, 127wrdind 14610 . . . 4 (𝑋 ∈ Word ((mCN‘𝑇) ∪ 𝑉) → (𝐹 ∈ ran 𝑆 → (ran (𝐹𝑋) ∩ 𝑉) = 𝑥 ∈ (ran 𝑋𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
129128com12 32 . . 3 (𝐹 ∈ ran 𝑆 → (𝑋 ∈ Word ((mCN‘𝑇) ∪ 𝑉) → (ran (𝐹𝑋) ∩ 𝑉) = 𝑥 ∈ (ran 𝑋𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
13010, 129sylbid 239 . 2 (𝐹 ∈ ran 𝑆 → (𝑋𝑅 → (ran (𝐹𝑋) ∩ 𝑉) = 𝑥 ∈ (ran 𝑋𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
131130imp 407 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 396   = wceq 1541  wcel 2106  Vcvv 3445  cdif 3907  cun 3908  cin 3909  wss 3910  c0 4282  {csn 4586   ciun 4954  ran crn 5634  wf 6492  cfv 6496  (class class class)co 7357  Word cword 14402   ++ cconcat 14458  ⟨“cs1 14483  mCNcmcn 34054  mVRcmvar 34055  mRExcmrex 34060  mRSubstcmrsub 34064
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-mrsub 34084
This theorem is referenced by:  msubvrs  34154
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