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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 4065 . . . . . 6 (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
2 mrsubco.s . . . . . . . . 9 𝑆 = (mRSubst‘𝑇)
3 fvprc 6325 . . . . . . . . 9 𝑇 ∈ V → (mRSubst‘𝑇) = ∅)
42, 3syl5eq 2815 . . . . . . . 8 𝑇 ∈ V → 𝑆 = ∅)
54rneqd 5490 . . . . . . 7 𝑇 ∈ V → ran 𝑆 = ran ∅)
6 rn0 5514 . . . . . . 7 ran ∅ = ∅
75, 6syl6eq 2819 . . . . . 6 𝑇 ∈ V → ran 𝑆 = ∅)
81, 7nsyl2 144 . . . . 5 (𝐹 ∈ ran 𝑆𝑇 ∈ V)
9 eqid 2769 . . . . . 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 2834 . . 3 (𝐹 ∈ ran 𝑆 → (𝑋𝑅𝑋 ∈ Word ((mCN‘𝑇) ∪ 𝑉)))
15 fveq2 6331 . . . . . . . . 9 (𝑣 = ∅ → (𝐹𝑣) = (𝐹‘∅))
1615rneqd 5490 . . . . . . . 8 (𝑣 = ∅ → ran (𝐹𝑣) = ran (𝐹‘∅))
1716ineq1d 3961 . . . . . . 7 (𝑣 = ∅ → (ran (𝐹𝑣) ∩ 𝑉) = (ran (𝐹‘∅) ∩ 𝑉))
18 rneq 5488 . . . . . . . . . . . 12 (𝑣 = ∅ → ran 𝑣 = ran ∅)
1918, 6syl6eq 2819 . . . . . . . . . . 11 (𝑣 = ∅ → ran 𝑣 = ∅)
2019ineq1d 3961 . . . . . . . . . 10 (𝑣 = ∅ → (ran 𝑣𝑉) = (∅ ∩ 𝑉))
21 0in 4110 . . . . . . . . . 10 (∅ ∩ 𝑉) = ∅
2220, 21syl6eq 2819 . . . . . . . . 9 (𝑣 = ∅ → (ran 𝑣𝑉) = ∅)
2322iuneq1d 4676 . . . . . . . 8 (𝑣 = ∅ → 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ ∅ (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
24 0iun 4708 . . . . . . . 8 𝑥 ∈ ∅ (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∅
2523, 24syl6eq 2819 . . . . . . 7 (𝑣 = ∅ → 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∅)
2617, 25eqeq12d 2784 . . . . . 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 6331 . . . . . . . . 9 (𝑣 = 𝑦 → (𝐹𝑣) = (𝐹𝑦))
2928rneqd 5490 . . . . . . . 8 (𝑣 = 𝑦 → ran (𝐹𝑣) = ran (𝐹𝑦))
3029ineq1d 3961 . . . . . . 7 (𝑣 = 𝑦 → (ran (𝐹𝑣) ∩ 𝑉) = (ran (𝐹𝑦) ∩ 𝑉))
31 rneq 5488 . . . . . . . . 9 (𝑣 = 𝑦 → ran 𝑣 = ran 𝑦)
3231ineq1d 3961 . . . . . . . 8 (𝑣 = 𝑦 → (ran 𝑣𝑉) = (ran 𝑦𝑉))
3332iuneq1d 4676 . . . . . . 7 (𝑣 = 𝑦 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
3430, 33eqeq12d 2784 . . . . . 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 6331 . . . . . . . . 9 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → (𝐹𝑣) = (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)))
3736rneqd 5490 . . . . . . . 8 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → ran (𝐹𝑣) = ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)))
3837ineq1d 3961 . . . . . . 7 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → (ran (𝐹𝑣) ∩ 𝑉) = (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉))
39 rneq 5488 . . . . . . . . 9 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → ran 𝑣 = ran (𝑦 ++ ⟨“𝑧”⟩))
4039ineq1d 3961 . . . . . . . 8 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → (ran 𝑣𝑉) = (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉))
4140iuneq1d 4676 . . . . . . 7 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
4238, 41eqeq12d 2784 . . . . . 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 6331 . . . . . . . . 9 (𝑣 = 𝑋 → (𝐹𝑣) = (𝐹𝑋))
4544rneqd 5490 . . . . . . . 8 (𝑣 = 𝑋 → ran (𝐹𝑣) = ran (𝐹𝑋))
4645ineq1d 3961 . . . . . . 7 (𝑣 = 𝑋 → (ran (𝐹𝑣) ∩ 𝑉) = (ran (𝐹𝑋) ∩ 𝑉))
47 rneq 5488 . . . . . . . . 9 (𝑣 = 𝑋 → ran 𝑣 = ran 𝑋)
4847ineq1d 3961 . . . . . . . 8 (𝑣 = 𝑋 → (ran 𝑣𝑉) = (ran 𝑋𝑉))
4948iuneq1d 4676 . . . . . . 7 (𝑣 = 𝑋 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ (ran 𝑋𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
5046, 49eqeq12d 2784 . . . . . 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 5490 . . . . . . . 8 (𝐹 ∈ ran 𝑆 → ran (𝐹‘∅) = ran ∅)
5453, 6syl6eq 2819 . . . . . . 7 (𝐹 ∈ ran 𝑆 → ran (𝐹‘∅) = ∅)
5554ineq1d 3961 . . . . . 6 (𝐹 ∈ ran 𝑆 → (ran (𝐹‘∅) ∩ 𝑉) = (∅ ∩ 𝑉))
5655, 21syl6eq 2819 . . . . 5 (𝐹 ∈ ran 𝑆 → (ran (𝐹‘∅) ∩ 𝑉) = ∅)
57 uneq1 3908 . . . . . . . 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 475 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝐹 ∈ ran 𝑆)
59 simprl 808 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉))
6013adantr 473 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
6159, 60eleqtrrd 2851 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑦𝑅)
62 simprr 810 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))
6362s1cld 13586 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ⟨“𝑧”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉))
6463, 60eleqtrrd 2851 . . . . . . . . . . . . . 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 1474 . . . . . . . . . . . . 13 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = ((𝐹𝑦) ++ (𝐹‘⟨“𝑧”⟩)))
6766rneqd 5490 . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝐹:𝑅𝑅)
7069, 61ffvelrnd 6502 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹𝑦) ∈ 𝑅)
7170, 60eleqtrd 2850 . . . . . . . . . . . . 13 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹𝑦) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
7269, 64ffvelrnd 6502 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘⟨“𝑧”⟩) ∈ 𝑅)
7372, 60eleqtrd 2850 . . . . . . . . . . . . 13 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘⟨“𝑧”⟩) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
74 ccatrn 13574 . . . . . . . . . . . . 13 (((𝐹𝑦) ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ (𝐹‘⟨“𝑧”⟩) ∈ Word ((mCN‘𝑇) ∪ 𝑉)) → ran ((𝐹𝑦) ++ (𝐹‘⟨“𝑧”⟩)) = (ran (𝐹𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)))
7571, 73, 74syl2anc 693 . . . . . . . . . . . 12 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran ((𝐹𝑦) ++ (𝐹‘⟨“𝑧”⟩)) = (ran (𝐹𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)))
7667, 75eqtrd 2803 . . . . . . . . . . 11 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = (ran (𝐹𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)))
7776ineq1d 3961 . . . . . . . . . 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 4021 . . . . . . . . . 10 ((ran (𝐹𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)) ∩ 𝑉) = ((ran (𝐹𝑦) ∩ 𝑉) ∪ (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
7977, 78syl6eq 2819 . . . . . . . . 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 13574 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ ⟨“𝑧”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉)) → ran (𝑦 ++ ⟨“𝑧”⟩) = (ran 𝑦 ∪ ran ⟨“𝑧”⟩))
8159, 63, 80syl2anc 693 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝑦 ++ ⟨“𝑧”⟩) = (ran 𝑦 ∪ ran ⟨“𝑧”⟩))
82 s1rn 13582 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉) → ran ⟨“𝑧”⟩ = {𝑧})
8382ad2antll 764 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran ⟨“𝑧”⟩ = {𝑧})
8483uneq2d 3915 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran 𝑦 ∪ ran ⟨“𝑧”⟩) = (ran 𝑦 ∪ {𝑧}))
8581, 84eqtrd 2803 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝑦 ++ ⟨“𝑧”⟩) = (ran 𝑦 ∪ {𝑧}))
8685ineq1d 3961 . . . . . . . . . . . . 13 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉) = ((ran 𝑦 ∪ {𝑧}) ∩ 𝑉))
87 indir 4021 . . . . . . . . . . . . 13 ((ran 𝑦 ∪ {𝑧}) ∩ 𝑉) = ((ran 𝑦𝑉) ∪ ({𝑧} ∩ 𝑉))
8886, 87syl6eq 2819 . . . . . . . . . . . 12 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉) = ((ran 𝑦𝑉) ∪ ({𝑧} ∩ 𝑉)))
8988iuneq1d 4676 . . . . . . . . . . 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 4736 . . . . . . . . . . 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 2819 . . . . . . . . . 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 480 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧𝑉) → 𝑧𝑉)
9392snssd 4472 . . . . . . . . . . . . . . 15 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧𝑉) → {𝑧} ⊆ 𝑉)
94 df-ss 3734 . . . . . . . . . . . . . . 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 4676 . . . . . . . . . . . . 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 3351 . . . . . . . . . . . . . 14 𝑧 ∈ V
98 s1eq 13583 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ⟨“𝑥”⟩ = ⟨“𝑧”⟩)
9998fveq2d 6335 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (𝐹‘⟨“𝑥”⟩) = (𝐹‘⟨“𝑧”⟩))
10099rneqd 5490 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → ran (𝐹‘⟨“𝑥”⟩) = ran (𝐹‘⟨“𝑧”⟩))
101100ineq1d 3961 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
10297, 101iunxsn 4734 . . . . . . . . . . . . 13 𝑥 ∈ {𝑧} (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉)
10396, 102syl6eq 2819 . . . . . . . . . . . 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 3953 . . . . . . . . . . . . . . 15 ({𝑧} ∩ 𝑉) = (𝑉 ∩ {𝑧})
105 simpr 480 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ¬ 𝑧𝑉)
106 disjsn 4380 . . . . . . . . . . . . . . . 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 2815 . . . . . . . . . . . . . 14 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ({𝑧} ∩ 𝑉) = ∅)
109108iuneq1d 4676 . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → 𝐹 ∈ ran 𝑆)
111 eldif 3730 . . . . . . . . . . . . . . . . . . . . 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 490 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → 𝑧 ∈ (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉))
114 difun2 4187 . . . . . . . . . . . . . . . . . . 19 (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉) = ((mCN‘𝑇) ∖ 𝑉)
115113, 114syl6eleq 2858 . . . . . . . . . . . . . . . . . 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 693 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → (𝐹‘⟨“𝑧”⟩) = ⟨“𝑧”⟩)
118117rneqd 5490 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ran (𝐹‘⟨“𝑧”⟩) = ran ⟨“𝑧”⟩)
11983adantr 473 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ran ⟨“𝑧”⟩ = {𝑧})
120118, 119eqtrd 2803 . . . . . . . . . . . . . . 15 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ran (𝐹‘⟨“𝑧”⟩) = {𝑧})
121120ineq1d 3961 . . . . . . . . . . . . . 14 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉) = ({𝑧} ∩ 𝑉))
122121, 108eqtrd 2803 . . . . . . . . . . . . 13 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉) = ∅)
12324, 109, 1223eqtr4a 2829 . . . . . . . . . . . 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 864 . . . . . . . . . . 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 3915 . . . . . . . . . 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 2803 . . . . . . . . 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 2784 . . . . . . . 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 449 . . . . . 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 13688 . . . 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 443 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 383   = wceq 1629  wcel 2143  Vcvv 3348  cdif 3717  cun 3718  cin 3719  wss 3720  c0 4060  {csn 4313   ciun 4651  ran crn 5249  wf 6026  cfv 6030  (class class class)co 6791  Word cword 13490   ++ cconcat 13492  ⟨“cs1 13493  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 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-rep 4901  ax-sep 4911  ax-nul 4919  ax-pow 4970  ax-pr 5033  ax-un 7094  ax-cnex 10192  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1070  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-nel 3045  df-ral 3064  df-rex 3065  df-reu 3066  df-rmo 3067  df-rab 3068  df-v 3350  df-sbc 3585  df-csb 3680  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-pss 3736  df-nul 4061  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4572  df-int 4609  df-iun 4653  df-br 4784  df-opab 4844  df-mpt 4861  df-tr 4884  df-id 5156  df-eprel 5161  df-po 5169  df-so 5170  df-fr 5207  df-we 5209  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-rn 5259  df-res 5260  df-ima 5261  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-om 7211  df-1st 7313  df-2nd 7314  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-oadd 7715  df-er 7894  df-map 8009  df-pm 8010  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-card 8963  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-nn 11221  df-2 11279  df-n0 11493  df-xnn0 11564  df-z 11578  df-uz 11888  df-fz 12533  df-fzo 12673  df-seq 13009  df-hash 13325  df-word 13498  df-lsw 13499  df-concat 13500  df-s1 13501  df-substr 13502  df-struct 16072  df-ndx 16073  df-slot 16074  df-base 16076  df-sets 16077  df-ress 16078  df-plusg 16168  df-0g 16316  df-gsum 16317  df-mgm 17456  df-sgrp 17498  df-mnd 17509  df-submnd 17550  df-frmd 17600  df-mrex 31722  df-mrsub 31726
This theorem is referenced by:  msubvrs  31796
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