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Theorem msubco 31133
 Description: The composition of two substitutions is a substitution. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypothesis
Ref Expression
msubco.s 𝑆 = (mSubst‘𝑇)
Assertion
Ref Expression
msubco ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → (𝐹𝐺) ∈ ran 𝑆)

Proof of Theorem msubco
Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . . . 5 (mEx‘𝑇) = (mEx‘𝑇)
2 eqid 2621 . . . . 5 (mRSubst‘𝑇) = (mRSubst‘𝑇)
3 msubco.s . . . . 5 𝑆 = (mSubst‘𝑇)
41, 2, 3elmsubrn 31130 . . . 4 ran 𝑆 = ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩))
54eleq2i 2690 . . 3 (𝐹 ∈ ran 𝑆𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩)))
6 eqid 2621 . . . 4 (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩)) = (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩))
7 fvex 6158 . . . . 5 (mEx‘𝑇) ∈ V
87mptex 6440 . . . 4 (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∈ V
96, 8elrnmpti 5336 . . 3 (𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩)) ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩))
105, 9bitri 264 . 2 (𝐹 ∈ ran 𝑆 ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩))
111, 2, 3elmsubrn 31130 . . . 4 ran 𝑆 = ran (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩))
1211eleq2i 2690 . . 3 (𝐺 ∈ ran 𝑆𝐺 ∈ ran (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)))
13 eqid 2621 . . . 4 (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) = (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩))
147mptex 6440 . . . 4 (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩) ∈ V
1513, 14elrnmpti 5336 . . 3 (𝐺 ∈ ran (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) ↔ ∃𝑔 ∈ ran (mRSubst‘𝑇)𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩))
1612, 15bitri 264 . 2 (𝐺 ∈ ran 𝑆 ↔ ∃𝑔 ∈ ran (mRSubst‘𝑇)𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩))
17 reeanv 3097 . . 3 (∃𝑓 ∈ ran (mRSubst‘𝑇)∃𝑔 ∈ ran (mRSubst‘𝑇)(𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∧ 𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) ↔ (∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∧ ∃𝑔 ∈ ran (mRSubst‘𝑇)𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)))
18 simpr 477 . . . . . . . . . . . 12 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑦 ∈ (mEx‘𝑇))
19 eqid 2621 . . . . . . . . . . . . 13 (mTC‘𝑇) = (mTC‘𝑇)
20 eqid 2621 . . . . . . . . . . . . 13 (mREx‘𝑇) = (mREx‘𝑇)
2119, 1, 20mexval 31104 . . . . . . . . . . . 12 (mEx‘𝑇) = ((mTC‘𝑇) × (mREx‘𝑇))
2218, 21syl6eleq 2708 . . . . . . . . . . 11 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑦 ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
23 xp1st 7143 . . . . . . . . . . 11 (𝑦 ∈ ((mTC‘𝑇) × (mREx‘𝑇)) → (1st𝑦) ∈ (mTC‘𝑇))
2422, 23syl 17 . . . . . . . . . 10 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → (1st𝑦) ∈ (mTC‘𝑇))
252, 20mrsubf 31119 . . . . . . . . . . . 12 (𝑔 ∈ ran (mRSubst‘𝑇) → 𝑔:(mREx‘𝑇)⟶(mREx‘𝑇))
2625ad2antlr 762 . . . . . . . . . . 11 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑔:(mREx‘𝑇)⟶(mREx‘𝑇))
27 xp2nd 7144 . . . . . . . . . . . 12 (𝑦 ∈ ((mTC‘𝑇) × (mREx‘𝑇)) → (2nd𝑦) ∈ (mREx‘𝑇))
2822, 27syl 17 . . . . . . . . . . 11 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → (2nd𝑦) ∈ (mREx‘𝑇))
2926, 28ffvelrnd 6316 . . . . . . . . . 10 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → (𝑔‘(2nd𝑦)) ∈ (mREx‘𝑇))
30 opelxpi 5108 . . . . . . . . . 10 (((1st𝑦) ∈ (mTC‘𝑇) ∧ (𝑔‘(2nd𝑦)) ∈ (mREx‘𝑇)) → ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
3124, 29, 30syl2anc 692 . . . . . . . . 9 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
3231, 21syl6eleqr 2709 . . . . . . . 8 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ ∈ (mEx‘𝑇))
33 eqidd 2622 . . . . . . . 8 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩) = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩))
34 eqidd 2622 . . . . . . . 8 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩))
35 fvex 6158 . . . . . . . . . 10 (1st𝑦) ∈ V
36 fvex 6158 . . . . . . . . . 10 (𝑔‘(2nd𝑦)) ∈ V
3735, 36op1std 7123 . . . . . . . . 9 (𝑥 = ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ → (1st𝑥) = (1st𝑦))
3835, 36op2ndd 7124 . . . . . . . . . 10 (𝑥 = ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ → (2nd𝑥) = (𝑔‘(2nd𝑦)))
3938fveq2d 6152 . . . . . . . . 9 (𝑥 = ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ → (𝑓‘(2nd𝑥)) = (𝑓‘(𝑔‘(2nd𝑦))))
4037, 39opeq12d 4378 . . . . . . . 8 (𝑥 = ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ → ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩ = ⟨(1st𝑦), (𝑓‘(𝑔‘(2nd𝑦)))⟩)
4132, 33, 34, 40fmptco 6351 . . . . . . 7 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∘ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑓‘(𝑔‘(2nd𝑦)))⟩))
42 fvco3 6232 . . . . . . . . . 10 ((𝑔:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ (2nd𝑦) ∈ (mREx‘𝑇)) → ((𝑓𝑔)‘(2nd𝑦)) = (𝑓‘(𝑔‘(2nd𝑦))))
4326, 28, 42syl2anc 692 . . . . . . . . 9 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ((𝑓𝑔)‘(2nd𝑦)) = (𝑓‘(𝑔‘(2nd𝑦))))
4443opeq2d 4377 . . . . . . . 8 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩ = ⟨(1st𝑦), (𝑓‘(𝑔‘(2nd𝑦)))⟩)
4544mpteq2dva 4704 . . . . . . 7 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩) = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑓‘(𝑔‘(2nd𝑦)))⟩))
4641, 45eqtr4d 2658 . . . . . 6 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∘ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩))
472mrsubco 31123 . . . . . . . 8 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑓𝑔) ∈ ran (mRSubst‘𝑇))
487mptex 6440 . . . . . . . 8 (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩) ∈ V
49 eqid 2621 . . . . . . . . 9 ( ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (‘(2nd𝑦))⟩)) = ( ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (‘(2nd𝑦))⟩))
50 fveq1 6147 . . . . . . . . . . 11 ( = (𝑓𝑔) → (‘(2nd𝑦)) = ((𝑓𝑔)‘(2nd𝑦)))
5150opeq2d 4377 . . . . . . . . . 10 ( = (𝑓𝑔) → ⟨(1st𝑦), (‘(2nd𝑦))⟩ = ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩)
5251mpteq2dv 4705 . . . . . . . . 9 ( = (𝑓𝑔) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (‘(2nd𝑦))⟩) = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩))
5349, 52elrnmpt1s 5333 . . . . . . . 8 (((𝑓𝑔) ∈ ran (mRSubst‘𝑇) ∧ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩) ∈ V) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩) ∈ ran ( ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (‘(2nd𝑦))⟩)))
5447, 48, 53sylancl 693 . . . . . . 7 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩) ∈ ran ( ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (‘(2nd𝑦))⟩)))
551, 2, 3elmsubrn 31130 . . . . . . 7 ran 𝑆 = ran ( ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (‘(2nd𝑦))⟩))
5654, 55syl6eleqr 2709 . . . . . 6 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩) ∈ ran 𝑆)
5746, 56eqeltrd 2698 . . . . 5 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∘ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) ∈ ran 𝑆)
58 coeq1 5239 . . . . . . 7 (𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) → (𝐹𝐺) = ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∘ 𝐺))
59 coeq2 5240 . . . . . . 7 (𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩) → ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∘ 𝐺) = ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∘ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)))
6058, 59sylan9eq 2675 . . . . . 6 ((𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∧ 𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) → (𝐹𝐺) = ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∘ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)))
6160eleq1d 2683 . . . . 5 ((𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∧ 𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) → ((𝐹𝐺) ∈ ran 𝑆 ↔ ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∘ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) ∈ ran 𝑆))
6257, 61syl5ibrcom 237 . . . 4 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → ((𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∧ 𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) → (𝐹𝐺) ∈ ran 𝑆))
6362rexlimivv 3029 . . 3 (∃𝑓 ∈ ran (mRSubst‘𝑇)∃𝑔 ∈ ran (mRSubst‘𝑇)(𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∧ 𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) → (𝐹𝐺) ∈ ran 𝑆)
6417, 63sylbir 225 . 2 ((∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∧ ∃𝑔 ∈ ran (mRSubst‘𝑇)𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) → (𝐹𝐺) ∈ ran 𝑆)
6510, 16, 64syl2anb 496 1 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → (𝐹𝐺) ∈ ran 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∃wrex 2908  Vcvv 3186  ⟨cop 4154   ↦ cmpt 4673   × cxp 5072  ran crn 5075   ∘ ccom 5078  ⟶wf 5843  ‘cfv 5847  1st c1st 7111  2nd c2nd 7112  mTCcmtc 31066  mRExcmrex 31068  mExcmex 31069  mRSubstcmrsub 31072  mSubstcmsub 31073 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-xnn0 11308  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-seq 12742  df-hash 13058  df-word 13238  df-lsw 13239  df-concat 13240  df-s1 13241  df-substr 13242  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-0g 16023  df-gsum 16024  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-submnd 17257  df-frmd 17307  df-vrmd 17308  df-mrex 31088  df-mex 31089  df-mrsub 31092  df-msub 31093 This theorem is referenced by:  mclsppslem  31185
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