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Theorem mrsubvrs 32672
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 4303 . . . . . 6 (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
2 mrsubco.s . . . . . . 7 𝑆 = (mRSubst‘𝑇)
32rnfvprc 6663 . . . . . 6 𝑇 ∈ V → ran 𝑆 = ∅)
41, 3nsyl2 143 . . . . 5 (𝐹 ∈ ran 𝑆𝑇 ∈ V)
5 eqid 2826 . . . . . 6 (mCN‘𝑇) = (mCN‘𝑇)
6 mrsubvrs.v . . . . . 6 𝑉 = (mVR‘𝑇)
7 mrsubvrs.r . . . . . 6 𝑅 = (mREx‘𝑇)
85, 6, 7mrexval 32651 . . . . 5 (𝑇 ∈ V → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
94, 8syl 17 . . . 4 (𝐹 ∈ ran 𝑆𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
109eleq2d 2903 . . 3 (𝐹 ∈ ran 𝑆 → (𝑋𝑅𝑋 ∈ Word ((mCN‘𝑇) ∪ 𝑉)))
11 fveq2 6669 . . . . . . . . 9 (𝑣 = ∅ → (𝐹𝑣) = (𝐹‘∅))
1211rneqd 5807 . . . . . . . 8 (𝑣 = ∅ → ran (𝐹𝑣) = ran (𝐹‘∅))
1312ineq1d 4192 . . . . . . 7 (𝑣 = ∅ → (ran (𝐹𝑣) ∩ 𝑉) = (ran (𝐹‘∅) ∩ 𝑉))
14 rneq 5805 . . . . . . . . . . . 12 (𝑣 = ∅ → ran 𝑣 = ran ∅)
15 rn0 5795 . . . . . . . . . . . 12 ran ∅ = ∅
1614, 15syl6eq 2877 . . . . . . . . . . 11 (𝑣 = ∅ → ran 𝑣 = ∅)
1716ineq1d 4192 . . . . . . . . . 10 (𝑣 = ∅ → (ran 𝑣𝑉) = (∅ ∩ 𝑉))
18 0in 4351 . . . . . . . . . 10 (∅ ∩ 𝑉) = ∅
1917, 18syl6eq 2877 . . . . . . . . 9 (𝑣 = ∅ → (ran 𝑣𝑉) = ∅)
2019iuneq1d 4943 . . . . . . . 8 (𝑣 = ∅ → 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ ∅ (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
21 0iun 4983 . . . . . . . 8 𝑥 ∈ ∅ (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∅
2220, 21syl6eq 2877 . . . . . . 7 (𝑣 = ∅ → 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∅)
2313, 22eqeq12d 2842 . . . . . 6 (𝑣 = ∅ → ((ran (𝐹𝑣) ∩ 𝑉) = 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ↔ (ran (𝐹‘∅) ∩ 𝑉) = ∅))
2423imbi2d 342 . . . . 5 (𝑣 = ∅ → ((𝐹 ∈ ran 𝑆 → (ran (𝐹𝑣) ∩ 𝑉) = 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)) ↔ (𝐹 ∈ ran 𝑆 → (ran (𝐹‘∅) ∩ 𝑉) = ∅)))
25 fveq2 6669 . . . . . . . . 9 (𝑣 = 𝑦 → (𝐹𝑣) = (𝐹𝑦))
2625rneqd 5807 . . . . . . . 8 (𝑣 = 𝑦 → ran (𝐹𝑣) = ran (𝐹𝑦))
2726ineq1d 4192 . . . . . . 7 (𝑣 = 𝑦 → (ran (𝐹𝑣) ∩ 𝑉) = (ran (𝐹𝑦) ∩ 𝑉))
28 rneq 5805 . . . . . . . . 9 (𝑣 = 𝑦 → ran 𝑣 = ran 𝑦)
2928ineq1d 4192 . . . . . . . 8 (𝑣 = 𝑦 → (ran 𝑣𝑉) = (ran 𝑦𝑉))
3029iuneq1d 4943 . . . . . . 7 (𝑣 = 𝑦 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
3127, 30eqeq12d 2842 . . . . . 6 (𝑣 = 𝑦 → ((ran (𝐹𝑣) ∩ 𝑉) = 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ↔ (ran (𝐹𝑦) ∩ 𝑉) = 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
3231imbi2d 342 . . . . 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 6669 . . . . . . . . 9 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → (𝐹𝑣) = (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)))
3433rneqd 5807 . . . . . . . 8 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → ran (𝐹𝑣) = ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)))
3534ineq1d 4192 . . . . . . 7 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → (ran (𝐹𝑣) ∩ 𝑉) = (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉))
36 rneq 5805 . . . . . . . . 9 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → ran 𝑣 = ran (𝑦 ++ ⟨“𝑧”⟩))
3736ineq1d 4192 . . . . . . . 8 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → (ran 𝑣𝑉) = (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉))
3837iuneq1d 4943 . . . . . . 7 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
3935, 38eqeq12d 2842 . . . . . 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 342 . . . . 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 6669 . . . . . . . . 9 (𝑣 = 𝑋 → (𝐹𝑣) = (𝐹𝑋))
4241rneqd 5807 . . . . . . . 8 (𝑣 = 𝑋 → ran (𝐹𝑣) = ran (𝐹𝑋))
4342ineq1d 4192 . . . . . . 7 (𝑣 = 𝑋 → (ran (𝐹𝑣) ∩ 𝑉) = (ran (𝐹𝑋) ∩ 𝑉))
44 rneq 5805 . . . . . . . . 9 (𝑣 = 𝑋 → ran 𝑣 = ran 𝑋)
4544ineq1d 4192 . . . . . . . 8 (𝑣 = 𝑋 → (ran 𝑣𝑉) = (ran 𝑋𝑉))
4645iuneq1d 4943 . . . . . . 7 (𝑣 = 𝑋 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ (ran 𝑋𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
4743, 46eqeq12d 2842 . . . . . 6 (𝑣 = 𝑋 → ((ran (𝐹𝑣) ∩ 𝑉) = 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ↔ (ran (𝐹𝑋) ∩ 𝑉) = 𝑥 ∈ (ran 𝑋𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
4847imbi2d 342 . . . . 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 32666 . . . . . . . . 9 (𝐹 ∈ ran 𝑆 → (𝐹‘∅) = ∅)
5049rneqd 5807 . . . . . . . 8 (𝐹 ∈ ran 𝑆 → ran (𝐹‘∅) = ran ∅)
5150, 15syl6eq 2877 . . . . . . 7 (𝐹 ∈ ran 𝑆 → ran (𝐹‘∅) = ∅)
5251ineq1d 4192 . . . . . 6 (𝐹 ∈ ran 𝑆 → (ran (𝐹‘∅) ∩ 𝑉) = (∅ ∩ 𝑉))
5352, 18syl6eq 2877 . . . . 5 (𝐹 ∈ ran 𝑆 → (ran (𝐹‘∅) ∩ 𝑉) = ∅)
54 uneq1 4136 . . . . . . . 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 767 . . . . . . . . . . . . . . 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 2921 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑦𝑅)
59 simprr 769 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))
6059s1cld 13952 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ⟨“𝑧”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉))
6160, 57eleqtrrd 2921 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ⟨“𝑧”⟩ ∈ 𝑅)
622, 7mrsubccat 32668 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆𝑦𝑅 ∧ ⟨“𝑧”⟩ ∈ 𝑅) → (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = ((𝐹𝑦) ++ (𝐹‘⟨“𝑧”⟩)))
6355, 58, 61, 62syl3anc 1365 . . . . . . . . . . . . 13 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = ((𝐹𝑦) ++ (𝐹‘⟨“𝑧”⟩)))
6463rneqd 5807 . . . . . . . . . . . 12 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = ran ((𝐹𝑦) ++ (𝐹‘⟨“𝑧”⟩)))
652, 7mrsubf 32667 . . . . . . . . . . . . . . . 16 (𝐹 ∈ ran 𝑆𝐹:𝑅𝑅)
6665adantr 481 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝐹:𝑅𝑅)
6766, 58ffvelrnd 6850 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹𝑦) ∈ 𝑅)
6867, 57eleqtrd 2920 . . . . . . . . . . . . 13 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹𝑦) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
6966, 61ffvelrnd 6850 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘⟨“𝑧”⟩) ∈ 𝑅)
7069, 57eleqtrd 2920 . . . . . . . . . . . . 13 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘⟨“𝑧”⟩) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
71 ccatrn 13938 . . . . . . . . . . . . 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 2861 . . . . . . . . . . 11 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = (ran (𝐹𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)))
7473ineq1d 4192 . . . . . . . . . 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 4256 . . . . . . . . . 10 ((ran (𝐹𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)) ∩ 𝑉) = ((ran (𝐹𝑦) ∩ 𝑉) ∪ (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
7674, 75syl6eq 2877 . . . . . . . . 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 13938 . . . . . . . . . . . . . . . 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 13948 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉) → ran ⟨“𝑧”⟩ = {𝑧})
8079ad2antll 725 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran ⟨“𝑧”⟩ = {𝑧})
8180uneq2d 4143 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran 𝑦 ∪ ran ⟨“𝑧”⟩) = (ran 𝑦 ∪ {𝑧}))
8278, 81eqtrd 2861 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝑦 ++ ⟨“𝑧”⟩) = (ran 𝑦 ∪ {𝑧}))
8382ineq1d 4192 . . . . . . . . . . . . 13 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉) = ((ran 𝑦 ∪ {𝑧}) ∩ 𝑉))
84 indir 4256 . . . . . . . . . . . . 13 ((ran 𝑦 ∪ {𝑧}) ∩ 𝑉) = ((ran 𝑦𝑉) ∪ ({𝑧} ∩ 𝑉))
8583, 84syl6eq 2877 . . . . . . . . . . . 12 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉) = ((ran 𝑦𝑉) ∪ ({𝑧} ∩ 𝑉)))
8685iuneq1d 4943 . . . . . . . . . . 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 5013 . . . . . . . . . . 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, 87syl6eq 2877 . . . . . . . . . 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 4741 . . . . . . . . . . . . . . 15 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧𝑉) → {𝑧} ⊆ 𝑉)
91 df-ss 3956 . . . . . . . . . . . . . . 15 ({𝑧} ⊆ 𝑉 ↔ ({𝑧} ∩ 𝑉) = {𝑧})
9290, 91sylib 219 . . . . . . . . . . . . . 14 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧𝑉) → ({𝑧} ∩ 𝑉) = {𝑧})
9392iuneq1d 4943 . . . . . . . . . . . . 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 3503 . . . . . . . . . . . . . 14 𝑧 ∈ V
95 s1eq 13949 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ⟨“𝑥”⟩ = ⟨“𝑧”⟩)
9695fveq2d 6673 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (𝐹‘⟨“𝑥”⟩) = (𝐹‘⟨“𝑧”⟩))
9796rneqd 5807 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → ran (𝐹‘⟨“𝑥”⟩) = ran (𝐹‘⟨“𝑧”⟩))
9897ineq1d 4192 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
9994, 98iunxsn 5010 . . . . . . . . . . . . 13 𝑥 ∈ {𝑧} (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉)
10093, 99syl6eq 2877 . . . . . . . . . . . 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 4182 . . . . . . . . . . . . . . 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 4646 . . . . . . . . . . . . . . . 16 ((𝑉 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑉)
104102, 103sylibr 235 . . . . . . . . . . . . . . 15 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → (𝑉 ∩ {𝑧}) = ∅)
105101, 104syl5eq 2873 . . . . . . . . . . . . . 14 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ({𝑧} ∩ 𝑉) = ∅)
106105iuneq1d 4943 . . . . . . . . . . . . 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 3950 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉) ↔ (𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉) ∧ ¬ 𝑧𝑉))
109108biimpri 229 . . . . . . . . . . . . . . . . . . . 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 4432 . . . . . . . . . . . . . . . . . . 19 (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉) = ((mCN‘𝑇) ∖ 𝑉)
112110, 111syl6eleq 2928 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → 𝑧 ∈ ((mCN‘𝑇) ∖ 𝑉))
1132, 7, 6, 5mrsubcn 32669 . . . . . . . . . . . . . . . . . 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 5807 . . . . . . . . . . . . . . . 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 2861 . . . . . . . . . . . . . . 15 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ran (𝐹‘⟨“𝑧”⟩) = {𝑧})
118117ineq1d 4192 . . . . . . . . . . . . . 14 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉) = ({𝑧} ∩ 𝑉))
119118, 105eqtrd 2861 . . . . . . . . . . . . 13 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉) = ∅)
12021, 106, 1193eqtr4a 2887 . . . . . . . . . . . 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 809 . . . . . . . . . . 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 4143 . . . . . . . . . 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 2861 . . . . . . . . 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 2842 . . . . . . . 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 247 . . . . . . 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 14079 . . . 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 241 . 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 1530  wcel 2107  Vcvv 3500  cdif 3937  cun 3938  cin 3939  wss 3940  c0 4295  {csn 4564   ciun 4917  ran crn 5555  wf 6350  cfv 6354  (class class class)co 7150  Word cword 13856   ++ cconcat 13917  ⟨“cs1 13944  mCNcmcn 32610  mVRcmvar 32611  mRExcmrex 32616  mRSubstcmrsub 32620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8284  df-map 8403  df-pm 8404  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-n0 11892  df-xnn0 11962  df-z 11976  df-uz 12238  df-fz 12888  df-fzo 13029  df-seq 13365  df-hash 13686  df-word 13857  df-lsw 13910  df-concat 13918  df-s1 13945  df-substr 13998  df-pfx 14028  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-0g 16710  df-gsum 16711  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-submnd 17952  df-frmd 18009  df-mrex 32636  df-mrsub 32640
This theorem is referenced by:  msubvrs  32710
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