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Theorem mrsubrn 32663
 Description: Although it is defined for partial mappings of variables, every partial substitution is a substitution on some complete mapping of the variables. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mrsubvr.v 𝑉 = (mVR‘𝑇)
mrsubvr.r 𝑅 = (mREx‘𝑇)
mrsubvr.s 𝑆 = (mRSubst‘𝑇)
Assertion
Ref Expression
mrsubrn ran 𝑆 = (𝑆 “ (𝑅m 𝑉))

Proof of Theorem mrsubrn
Dummy variables 𝑒 𝑓 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mrsubvr.v . . . . . . 7 𝑉 = (mVR‘𝑇)
2 mrsubvr.r . . . . . . 7 𝑅 = (mREx‘𝑇)
3 mrsubvr.s . . . . . . 7 𝑆 = (mRSubst‘𝑇)
41, 2, 3mrsubff 32662 . . . . . 6 (𝑇 ∈ V → 𝑆:(𝑅pm 𝑉)⟶(𝑅m 𝑅))
54ffnd 6514 . . . . 5 (𝑇 ∈ V → 𝑆 Fn (𝑅pm 𝑉))
6 eleq1w 2900 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑣 → (𝑥 ∈ dom 𝑓𝑣 ∈ dom 𝑓))
7 fveq2 6669 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑣 → (𝑓𝑥) = (𝑓𝑣))
8 s1eq 13949 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑣 → ⟨“𝑥”⟩ = ⟨“𝑣”⟩)
96, 7, 8ifbieq12d 4497 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑣 → if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩) = if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩))
10 eqid 2826 . . . . . . . . . . . . . . . . 17 (𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩)) = (𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))
11 fvex 6682 . . . . . . . . . . . . . . . . . 18 (𝑓𝑣) ∈ V
12 s1cli 13954 . . . . . . . . . . . . . . . . . . 19 ⟨“𝑣”⟩ ∈ Word V
1312elexi 3519 . . . . . . . . . . . . . . . . . 18 ⟨“𝑣”⟩ ∈ V
1411, 13ifex 4518 . . . . . . . . . . . . . . . . 17 if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩) ∈ V
159, 10, 14fvmpt 6767 . . . . . . . . . . . . . . . 16 (𝑣𝑉 → ((𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))‘𝑣) = if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩))
1615adantl 482 . . . . . . . . . . . . . . 15 (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑣𝑉) → ((𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))‘𝑣) = if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩))
1716ifeq1da 4500 . . . . . . . . . . . . . 14 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → if(𝑣𝑉, ((𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))‘𝑣), ⟨“𝑣”⟩) = if(𝑣𝑉, if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩), ⟨“𝑣”⟩))
18 ifan 4521 . . . . . . . . . . . . . 14 if((𝑣𝑉𝑣 ∈ dom 𝑓), (𝑓𝑣), ⟨“𝑣”⟩) = if(𝑣𝑉, if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩), ⟨“𝑣”⟩)
1917, 18syl6eqr 2879 . . . . . . . . . . . . 13 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → if(𝑣𝑉, ((𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))‘𝑣), ⟨“𝑣”⟩) = if((𝑣𝑉𝑣 ∈ dom 𝑓), (𝑓𝑣), ⟨“𝑣”⟩))
20 elpmi 8420 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ (𝑅pm 𝑉) → (𝑓:dom 𝑓𝑅 ∧ dom 𝑓𝑉))
2120adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑓:dom 𝑓𝑅 ∧ dom 𝑓𝑉))
2221simprd 496 . . . . . . . . . . . . . . . . 17 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → dom 𝑓𝑉)
2322sseld 3970 . . . . . . . . . . . . . . . 16 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑣 ∈ dom 𝑓𝑣𝑉))
2423pm4.71rd 563 . . . . . . . . . . . . . . 15 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑣 ∈ dom 𝑓 ↔ (𝑣𝑉𝑣 ∈ dom 𝑓)))
2524bicomd 224 . . . . . . . . . . . . . 14 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ((𝑣𝑉𝑣 ∈ dom 𝑓) ↔ 𝑣 ∈ dom 𝑓))
2625ifbid 4492 . . . . . . . . . . . . 13 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → if((𝑣𝑉𝑣 ∈ dom 𝑓), (𝑓𝑣), ⟨“𝑣”⟩) = if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩))
2719, 26eqtr2d 2862 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩) = if(𝑣𝑉, ((𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))‘𝑣), ⟨“𝑣”⟩))
2827mpteq2dv 5159 . . . . . . . . . . 11 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) = (𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣𝑉, ((𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))‘𝑣), ⟨“𝑣”⟩)))
2928coeq1d 5731 . . . . . . . . . 10 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒) = ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣𝑉, ((𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))
3029oveq2d 7166 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)) = ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣𝑉, ((𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))
3130mpteq2dv 5159 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) = (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣𝑉, ((𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
32 eqid 2826 . . . . . . . . . 10 (mCN‘𝑇) = (mCN‘𝑇)
33 eqid 2826 . . . . . . . . . 10 (freeMnd‘((mCN‘𝑇) ∪ 𝑉)) = (freeMnd‘((mCN‘𝑇) ∪ 𝑉))
3432, 1, 2, 3, 33mrsubfval 32658 . . . . . . . . 9 ((𝑓:dom 𝑓𝑅 ∧ dom 𝑓𝑉) → (𝑆𝑓) = (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
3521, 34syl 17 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑆𝑓) = (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
3621simpld 495 . . . . . . . . . . . . 13 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → 𝑓:dom 𝑓𝑅)
3736adantr 481 . . . . . . . . . . . 12 (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑥𝑉) → 𝑓:dom 𝑓𝑅)
3837ffvelrnda 6849 . . . . . . . . . . 11 ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑥𝑉) ∧ 𝑥 ∈ dom 𝑓) → (𝑓𝑥) ∈ 𝑅)
39 elun2 4157 . . . . . . . . . . . . . 14 (𝑥𝑉𝑥 ∈ ((mCN‘𝑇) ∪ 𝑉))
4039ad2antlr 723 . . . . . . . . . . . . 13 ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑥𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → 𝑥 ∈ ((mCN‘𝑇) ∪ 𝑉))
4140s1cld 13952 . . . . . . . . . . . 12 ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑥𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → ⟨“𝑥”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉))
4232, 1, 2mrexval 32651 . . . . . . . . . . . . 13 (𝑇 ∈ V → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
4342ad3antrrr 726 . . . . . . . . . . . 12 ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑥𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
4441, 43eleqtrrd 2921 . . . . . . . . . . 11 ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑥𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → ⟨“𝑥”⟩ ∈ 𝑅)
4538, 44ifclda 4504 . . . . . . . . . 10 (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑥𝑉) → if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩) ∈ 𝑅)
4645fmpttd 6877 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩)):𝑉𝑅)
47 ssid 3993 . . . . . . . . 9 𝑉𝑉
4832, 1, 2, 3, 33mrsubfval 32658 . . . . . . . . 9 (((𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩)):𝑉𝑅𝑉𝑉) → (𝑆‘(𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))) = (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣𝑉, ((𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
4946, 47, 48sylancl 586 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑆‘(𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))) = (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣𝑉, ((𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
5031, 35, 493eqtr4d 2871 . . . . . . 7 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑆𝑓) = (𝑆‘(𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))))
515adantr 481 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → 𝑆 Fn (𝑅pm 𝑉))
52 mapsspm 8435 . . . . . . . . 9 (𝑅m 𝑉) ⊆ (𝑅pm 𝑉)
5352a1i 11 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑅m 𝑉) ⊆ (𝑅pm 𝑉))
542fvexi 6683 . . . . . . . . . 10 𝑅 ∈ V
551fvexi 6683 . . . . . . . . . 10 𝑉 ∈ V
5654, 55elmap 8430 . . . . . . . . 9 ((𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩)) ∈ (𝑅m 𝑉) ↔ (𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩)):𝑉𝑅)
5746, 56sylibr 235 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩)) ∈ (𝑅m 𝑉))
58 fnfvima 6991 . . . . . . . 8 ((𝑆 Fn (𝑅pm 𝑉) ∧ (𝑅m 𝑉) ⊆ (𝑅pm 𝑉) ∧ (𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩)) ∈ (𝑅m 𝑉)) → (𝑆‘(𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))) ∈ (𝑆 “ (𝑅m 𝑉)))
5951, 53, 57, 58syl3anc 1365 . . . . . . 7 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑆‘(𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))) ∈ (𝑆 “ (𝑅m 𝑉)))
6050, 59eqeltrd 2918 . . . . . 6 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑆𝑓) ∈ (𝑆 “ (𝑅m 𝑉)))
6160ralrimiva 3187 . . . . 5 (𝑇 ∈ V → ∀𝑓 ∈ (𝑅pm 𝑉)(𝑆𝑓) ∈ (𝑆 “ (𝑅m 𝑉)))
62 ffnfv 6880 . . . . 5 (𝑆:(𝑅pm 𝑉)⟶(𝑆 “ (𝑅m 𝑉)) ↔ (𝑆 Fn (𝑅pm 𝑉) ∧ ∀𝑓 ∈ (𝑅pm 𝑉)(𝑆𝑓) ∈ (𝑆 “ (𝑅m 𝑉))))
635, 61, 62sylanbrc 583 . . . 4 (𝑇 ∈ V → 𝑆:(𝑅pm 𝑉)⟶(𝑆 “ (𝑅m 𝑉)))
6463frnd 6520 . . 3 (𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅m 𝑉)))
653rnfvprc 6663 . . . 4 𝑇 ∈ V → ran 𝑆 = ∅)
66 0ss 4354 . . . 4 ∅ ⊆ (𝑆 “ (𝑅m 𝑉))
6765, 66eqsstrdi 4025 . . 3 𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅m 𝑉)))
6864, 67pm2.61i 183 . 2 ran 𝑆 ⊆ (𝑆 “ (𝑅m 𝑉))
69 imassrn 5939 . 2 (𝑆 “ (𝑅m 𝑉)) ⊆ ran 𝑆
7068, 69eqssi 3987 1 ran 𝑆 = (𝑆 “ (𝑅m 𝑉))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1530   ∈ wcel 2107  ∀wral 3143  Vcvv 3500   ∪ cun 3938   ⊆ wss 3940  ∅c0 4295  ifcif 4470   ↦ cmpt 5143  dom cdm 5554  ran crn 5555   “ cima 5557   ∘ ccom 5558   Fn wfn 6349  ⟶wf 6350  ‘cfv 6354  (class class class)co 7150   ↑m cmap 8401   ↑pm cpm 8402  Word cword 13856  ⟨“cs1 13944   Σg cgsu 16709  freeMndcfrmd 18007  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-z 11976  df-uz 12238  df-fz 12888  df-fzo 13029  df-seq 13365  df-hash 13686  df-word 13857  df-concat 13918  df-s1 13945  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:  mrsubff1o  32665  mrsub0  32666  mrsubccat  32668  mrsubcn  32669  msubrn  32679
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