Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mrsubvrs Structured version   Visualization version   GIF version

Theorem mrsubvrs 34450
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 4331 . . . . . 6 (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
2 mrsubco.s . . . . . . 7 𝑆 = (mRSubst‘𝑇)
32rnfvprc 6881 . . . . . 6 𝑇 ∈ V → ran 𝑆 = ∅)
41, 3nsyl2 141 . . . . 5 (𝐹 ∈ ran 𝑆𝑇 ∈ V)
5 eqid 2733 . . . . . 6 (mCN‘𝑇) = (mCN‘𝑇)
6 mrsubvrs.v . . . . . 6 𝑉 = (mVR‘𝑇)
7 mrsubvrs.r . . . . . 6 𝑅 = (mREx‘𝑇)
85, 6, 7mrexval 34429 . . . . 5 (𝑇 ∈ V → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
94, 8syl 17 . . . 4 (𝐹 ∈ ran 𝑆𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
109eleq2d 2820 . . 3 (𝐹 ∈ ran 𝑆 → (𝑋𝑅𝑋 ∈ Word ((mCN‘𝑇) ∪ 𝑉)))
11 fveq2 6887 . . . . . . . . 9 (𝑣 = ∅ → (𝐹𝑣) = (𝐹‘∅))
1211rneqd 5934 . . . . . . . 8 (𝑣 = ∅ → ran (𝐹𝑣) = ran (𝐹‘∅))
1312ineq1d 4209 . . . . . . 7 (𝑣 = ∅ → (ran (𝐹𝑣) ∩ 𝑉) = (ran (𝐹‘∅) ∩ 𝑉))
14 rneq 5932 . . . . . . . . . . . 12 (𝑣 = ∅ → ran 𝑣 = ran ∅)
15 rn0 5922 . . . . . . . . . . . 12 ran ∅ = ∅
1614, 15eqtrdi 2789 . . . . . . . . . . 11 (𝑣 = ∅ → ran 𝑣 = ∅)
1716ineq1d 4209 . . . . . . . . . 10 (𝑣 = ∅ → (ran 𝑣𝑉) = (∅ ∩ 𝑉))
18 0in 4391 . . . . . . . . . 10 (∅ ∩ 𝑉) = ∅
1917, 18eqtrdi 2789 . . . . . . . . 9 (𝑣 = ∅ → (ran 𝑣𝑉) = ∅)
2019iuneq1d 5022 . . . . . . . 8 (𝑣 = ∅ → 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ ∅ (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
21 0iun 5064 . . . . . . . 8 𝑥 ∈ ∅ (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∅
2220, 21eqtrdi 2789 . . . . . . 7 (𝑣 = ∅ → 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∅)
2313, 22eqeq12d 2749 . . . . . 6 (𝑣 = ∅ → ((ran (𝐹𝑣) ∩ 𝑉) = 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ↔ (ran (𝐹‘∅) ∩ 𝑉) = ∅))
2423imbi2d 341 . . . . 5 (𝑣 = ∅ → ((𝐹 ∈ ran 𝑆 → (ran (𝐹𝑣) ∩ 𝑉) = 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)) ↔ (𝐹 ∈ ran 𝑆 → (ran (𝐹‘∅) ∩ 𝑉) = ∅)))
25 fveq2 6887 . . . . . . . . 9 (𝑣 = 𝑦 → (𝐹𝑣) = (𝐹𝑦))
2625rneqd 5934 . . . . . . . 8 (𝑣 = 𝑦 → ran (𝐹𝑣) = ran (𝐹𝑦))
2726ineq1d 4209 . . . . . . 7 (𝑣 = 𝑦 → (ran (𝐹𝑣) ∩ 𝑉) = (ran (𝐹𝑦) ∩ 𝑉))
28 rneq 5932 . . . . . . . . 9 (𝑣 = 𝑦 → ran 𝑣 = ran 𝑦)
2928ineq1d 4209 . . . . . . . 8 (𝑣 = 𝑦 → (ran 𝑣𝑉) = (ran 𝑦𝑉))
3029iuneq1d 5022 . . . . . . 7 (𝑣 = 𝑦 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
3127, 30eqeq12d 2749 . . . . . 6 (𝑣 = 𝑦 → ((ran (𝐹𝑣) ∩ 𝑉) = 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ↔ (ran (𝐹𝑦) ∩ 𝑉) = 𝑥 ∈ (ran 𝑦𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
3231imbi2d 341 . . . . 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 6887 . . . . . . . . 9 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → (𝐹𝑣) = (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)))
3433rneqd 5934 . . . . . . . 8 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → ran (𝐹𝑣) = ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)))
3534ineq1d 4209 . . . . . . 7 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → (ran (𝐹𝑣) ∩ 𝑉) = (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉))
36 rneq 5932 . . . . . . . . 9 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → ran 𝑣 = ran (𝑦 ++ ⟨“𝑧”⟩))
3736ineq1d 4209 . . . . . . . 8 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → (ran 𝑣𝑉) = (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉))
3837iuneq1d 5022 . . . . . . 7 (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
3935, 38eqeq12d 2749 . . . . . 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 341 . . . . 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 6887 . . . . . . . . 9 (𝑣 = 𝑋 → (𝐹𝑣) = (𝐹𝑋))
4241rneqd 5934 . . . . . . . 8 (𝑣 = 𝑋 → ran (𝐹𝑣) = ran (𝐹𝑋))
4342ineq1d 4209 . . . . . . 7 (𝑣 = 𝑋 → (ran (𝐹𝑣) ∩ 𝑉) = (ran (𝐹𝑋) ∩ 𝑉))
44 rneq 5932 . . . . . . . . 9 (𝑣 = 𝑋 → ran 𝑣 = ran 𝑋)
4544ineq1d 4209 . . . . . . . 8 (𝑣 = 𝑋 → (ran 𝑣𝑉) = (ran 𝑋𝑉))
4645iuneq1d 5022 . . . . . . 7 (𝑣 = 𝑋 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = 𝑥 ∈ (ran 𝑋𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
4743, 46eqeq12d 2749 . . . . . 6 (𝑣 = 𝑋 → ((ran (𝐹𝑣) ∩ 𝑉) = 𝑥 ∈ (ran 𝑣𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ↔ (ran (𝐹𝑋) ∩ 𝑉) = 𝑥 ∈ (ran 𝑋𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
4847imbi2d 341 . . . . 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 34444 . . . . . . . . 9 (𝐹 ∈ ran 𝑆 → (𝐹‘∅) = ∅)
5049rneqd 5934 . . . . . . . 8 (𝐹 ∈ ran 𝑆 → ran (𝐹‘∅) = ran ∅)
5150, 15eqtrdi 2789 . . . . . . 7 (𝐹 ∈ ran 𝑆 → ran (𝐹‘∅) = ∅)
5251ineq1d 4209 . . . . . 6 (𝐹 ∈ ran 𝑆 → (ran (𝐹‘∅) ∩ 𝑉) = (∅ ∩ 𝑉))
5352, 18eqtrdi 2789 . . . . 5 (𝐹 ∈ ran 𝑆 → (ran (𝐹‘∅) ∩ 𝑉) = ∅)
54 uneq1 4154 . . . . . . . 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 484 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝐹 ∈ ran 𝑆)
56 simprl 770 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉))
579adantr 482 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
5856, 57eleqtrrd 2837 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑦𝑅)
59 simprr 772 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))
6059s1cld 14548 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ⟨“𝑧”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉))
6160, 57eleqtrrd 2837 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ⟨“𝑧”⟩ ∈ 𝑅)
622, 7mrsubccat 34446 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆𝑦𝑅 ∧ ⟨“𝑧”⟩ ∈ 𝑅) → (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = ((𝐹𝑦) ++ (𝐹‘⟨“𝑧”⟩)))
6355, 58, 61, 62syl3anc 1372 . . . . . . . . . . . . 13 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = ((𝐹𝑦) ++ (𝐹‘⟨“𝑧”⟩)))
6463rneqd 5934 . . . . . . . . . . . 12 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = ran ((𝐹𝑦) ++ (𝐹‘⟨“𝑧”⟩)))
652, 7mrsubf 34445 . . . . . . . . . . . . . . . 16 (𝐹 ∈ ran 𝑆𝐹:𝑅𝑅)
6665adantr 482 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝐹:𝑅𝑅)
6766, 58ffvelcdmd 7082 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹𝑦) ∈ 𝑅)
6867, 57eleqtrd 2836 . . . . . . . . . . . . 13 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹𝑦) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
6966, 61ffvelcdmd 7082 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘⟨“𝑧”⟩) ∈ 𝑅)
7069, 57eleqtrd 2836 . . . . . . . . . . . . 13 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘⟨“𝑧”⟩) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
71 ccatrn 14534 . . . . . . . . . . . . 13 (((𝐹𝑦) ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ (𝐹‘⟨“𝑧”⟩) ∈ Word ((mCN‘𝑇) ∪ 𝑉)) → ran ((𝐹𝑦) ++ (𝐹‘⟨“𝑧”⟩)) = (ran (𝐹𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)))
7268, 70, 71syl2anc 585 . . . . . . . . . . . 12 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran ((𝐹𝑦) ++ (𝐹‘⟨“𝑧”⟩)) = (ran (𝐹𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)))
7364, 72eqtrd 2773 . . . . . . . . . . 11 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = (ran (𝐹𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)))
7473ineq1d 4209 . . . . . . . . . 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 4273 . . . . . . . . . 10 ((ran (𝐹𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)) ∩ 𝑉) = ((ran (𝐹𝑦) ∩ 𝑉) ∪ (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
7674, 75eqtrdi 2789 . . . . . . . . 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 14534 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ ⟨“𝑧”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉)) → ran (𝑦 ++ ⟨“𝑧”⟩) = (ran 𝑦 ∪ ran ⟨“𝑧”⟩))
7856, 60, 77syl2anc 585 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝑦 ++ ⟨“𝑧”⟩) = (ran 𝑦 ∪ ran ⟨“𝑧”⟩))
79 s1rn 14544 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉) → ran ⟨“𝑧”⟩ = {𝑧})
8079ad2antll 728 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran ⟨“𝑧”⟩ = {𝑧})
8180uneq2d 4161 . . . . . . . . . . . . . . 15 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran 𝑦 ∪ ran ⟨“𝑧”⟩) = (ran 𝑦 ∪ {𝑧}))
8278, 81eqtrd 2773 . . . . . . . . . . . . . 14 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝑦 ++ ⟨“𝑧”⟩) = (ran 𝑦 ∪ {𝑧}))
8382ineq1d 4209 . . . . . . . . . . . . 13 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉) = ((ran 𝑦 ∪ {𝑧}) ∩ 𝑉))
84 indir 4273 . . . . . . . . . . . . 13 ((ran 𝑦 ∪ {𝑧}) ∩ 𝑉) = ((ran 𝑦𝑉) ∪ ({𝑧} ∩ 𝑉))
8583, 84eqtrdi 2789 . . . . . . . . . . . 12 ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉) = ((ran 𝑦𝑉) ∪ ({𝑧} ∩ 𝑉)))
8685iuneq1d 5022 . . . . . . . . . . 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 5095 . . . . . . . . . . 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 2789 . . . . . . . . . 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 486 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧𝑉) → 𝑧𝑉)
9089snssd 4810 . . . . . . . . . . . . . . 15 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧𝑉) → {𝑧} ⊆ 𝑉)
91 df-ss 3963 . . . . . . . . . . . . . . 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 5022 . . . . . . . . . . . . 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 3479 . . . . . . . . . . . . . 14 𝑧 ∈ V
95 s1eq 14545 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ⟨“𝑥”⟩ = ⟨“𝑧”⟩)
9695fveq2d 6891 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (𝐹‘⟨“𝑥”⟩) = (𝐹‘⟨“𝑧”⟩))
9796rneqd 5934 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → ran (𝐹‘⟨“𝑥”⟩) = ran (𝐹‘⟨“𝑧”⟩))
9897ineq1d 4209 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
9994, 98iunxsn 5092 . . . . . . . . . . . . 13 𝑥 ∈ {𝑧} (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉)
10093, 99eqtrdi 2789 . . . . . . . . . . . 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 4199 . . . . . . . . . . . . . . 15 ({𝑧} ∩ 𝑉) = (𝑉 ∩ {𝑧})
102 simpr 486 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ¬ 𝑧𝑉)
103 disjsn 4713 . . . . . . . . . . . . . . . 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 2785 . . . . . . . . . . . . . 14 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ({𝑧} ∩ 𝑉) = ∅)
106105iuneq1d 5022 . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → 𝐹 ∈ ran 𝑆)
108 eldif 3956 . . . . . . . . . . . . . . . . . . . . 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 581 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → 𝑧 ∈ (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉))
111 difun2 4478 . . . . . . . . . . . . . . . . . . 19 (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉) = ((mCN‘𝑇) ∖ 𝑉)
112110, 111eleqtrdi 2844 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → 𝑧 ∈ ((mCN‘𝑇) ∖ 𝑉))
1132, 7, 6, 5mrsubcn 34447 . . . . . . . . . . . . . . . . . 18 ((𝐹 ∈ ran 𝑆𝑧 ∈ ((mCN‘𝑇) ∖ 𝑉)) → (𝐹‘⟨“𝑧”⟩) = ⟨“𝑧”⟩)
114107, 112, 113syl2anc 585 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → (𝐹‘⟨“𝑧”⟩) = ⟨“𝑧”⟩)
115114rneqd 5934 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ran (𝐹‘⟨“𝑧”⟩) = ran ⟨“𝑧”⟩)
11680adantr 482 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ran ⟨“𝑧”⟩ = {𝑧})
117115, 116eqtrd 2773 . . . . . . . . . . . . . . 15 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → ran (𝐹‘⟨“𝑧”⟩) = {𝑧})
118117ineq1d 4209 . . . . . . . . . . . . . 14 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉) = ({𝑧} ∩ 𝑉))
119118, 105eqtrd 2773 . . . . . . . . . . . . 13 (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧𝑉) → (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉) = ∅)
12021, 106, 1193eqtr4a 2799 . . . . . . . . . . . 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 812 . . . . . . . . . . 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 4161 . . . . . . . . . 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 2773 . . . . . . . . 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 2749 . . . . . . . 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, 124imbitrrid 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 415 . . . . . 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 14667 . . . 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 408 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 397   = wceq 1542  wcel 2107  Vcvv 3475  cdif 3943  cun 3944  cin 3945  wss 3946  c0 4320  {csn 4626   ciun 4995  ran crn 5675  wf 6535  cfv 6539  (class class class)co 7403  Word cword 14459   ++ cconcat 14515  ⟨“cs1 14540  mCNcmcn 34388  mVRcmvar 34389  mRExcmrex 34394  mRSubstcmrsub 34398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5283  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-int 4949  df-iun 4997  df-br 5147  df-opab 5209  df-mpt 5230  df-tr 5264  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6296  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7359  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8260  df-wrecs 8291  df-recs 8365  df-rdg 8404  df-1o 8460  df-er 8698  df-map 8817  df-pm 8818  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-card 9929  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11441  df-neg 11442  df-nn 12208  df-2 12270  df-n0 12468  df-xnn0 12540  df-z 12554  df-uz 12818  df-fz 13480  df-fzo 13623  df-seq 13962  df-hash 14286  df-word 14460  df-lsw 14508  df-concat 14516  df-s1 14541  df-substr 14586  df-pfx 14616  df-struct 17075  df-sets 17092  df-slot 17110  df-ndx 17122  df-base 17140  df-ress 17169  df-plusg 17205  df-0g 17382  df-gsum 17383  df-mgm 18556  df-sgrp 18605  df-mnd 18621  df-submnd 18667  df-frmd 18725  df-mrex 34414  df-mrsub 34418
This theorem is referenced by:  msubvrs  34488
  Copyright terms: Public domain W3C validator