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Theorem mrsubrn 34493
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 34492 . . . . . 6 (𝑇 ∈ V → 𝑆:(𝑅pm 𝑉)⟶(𝑅m 𝑅))
54ffnd 6716 . . . . 5 (𝑇 ∈ V → 𝑆 Fn (𝑅pm 𝑉))
6 eleq1w 2817 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑣 → (𝑥 ∈ dom 𝑓𝑣 ∈ dom 𝑓))
7 fveq2 6889 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑣 → (𝑓𝑥) = (𝑓𝑣))
8 s1eq 14547 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑣 → ⟨“𝑥”⟩ = ⟨“𝑣”⟩)
96, 7, 8ifbieq12d 4556 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑣 → if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩) = if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩))
10 eqid 2733 . . . . . . . . . . . . . . . . 17 (𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩)) = (𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))
11 fvex 6902 . . . . . . . . . . . . . . . . . 18 (𝑓𝑣) ∈ V
12 s1cli 14552 . . . . . . . . . . . . . . . . . . 19 ⟨“𝑣”⟩ ∈ Word V
1312elexi 3494 . . . . . . . . . . . . . . . . . 18 ⟨“𝑣”⟩ ∈ V
1411, 13ifex 4578 . . . . . . . . . . . . . . . . 17 if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩) ∈ V
159, 10, 14fvmpt 6996 . . . . . . . . . . . . . . . 16 (𝑣𝑉 → ((𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))‘𝑣) = if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩))
1615adantl 483 . . . . . . . . . . . . . . 15 (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑣𝑉) → ((𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))‘𝑣) = if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩))
1716ifeq1da 4559 . . . . . . . . . . . . . 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 4581 . . . . . . . . . . . . . 14 if((𝑣𝑉𝑣 ∈ dom 𝑓), (𝑓𝑣), ⟨“𝑣”⟩) = if(𝑣𝑉, if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩), ⟨“𝑣”⟩)
1917, 18eqtr4di 2791 . . . . . . . . . . . . 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 8837 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ (𝑅pm 𝑉) → (𝑓:dom 𝑓𝑅 ∧ dom 𝑓𝑉))
2120adantl 483 . . . . . . . . . . . . . . . . . 18 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑓:dom 𝑓𝑅 ∧ dom 𝑓𝑉))
2221simprd 497 . . . . . . . . . . . . . . . . 17 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → dom 𝑓𝑉)
2322sseld 3981 . . . . . . . . . . . . . . . 16 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑣 ∈ dom 𝑓𝑣𝑉))
2423pm4.71rd 564 . . . . . . . . . . . . . . 15 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑣 ∈ dom 𝑓 ↔ (𝑣𝑉𝑣 ∈ dom 𝑓)))
2524bicomd 222 . . . . . . . . . . . . . 14 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ((𝑣𝑉𝑣 ∈ dom 𝑓) ↔ 𝑣 ∈ dom 𝑓))
2625ifbid 4551 . . . . . . . . . . . . 13 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → if((𝑣𝑉𝑣 ∈ dom 𝑓), (𝑓𝑣), ⟨“𝑣”⟩) = if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩))
2719, 26eqtr2d 2774 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩) = if(𝑣𝑉, ((𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))‘𝑣), ⟨“𝑣”⟩))
2827mpteq2dv 5250 . . . . . . . . . . 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 5860 . . . . . . . . . 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 7422 . . . . . . . . 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 5250 . . . . . . . 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 2733 . . . . . . . . . 10 (mCN‘𝑇) = (mCN‘𝑇)
33 eqid 2733 . . . . . . . . . 10 (freeMnd‘((mCN‘𝑇) ∪ 𝑉)) = (freeMnd‘((mCN‘𝑇) ∪ 𝑉))
3432, 1, 2, 3, 33mrsubfval 34488 . . . . . . . . 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 496 . . . . . . . . . . . . 13 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → 𝑓:dom 𝑓𝑅)
3736adantr 482 . . . . . . . . . . . 12 (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑥𝑉) → 𝑓:dom 𝑓𝑅)
3837ffvelcdmda 7084 . . . . . . . . . . 11 ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑥𝑉) ∧ 𝑥 ∈ dom 𝑓) → (𝑓𝑥) ∈ 𝑅)
39 elun2 4177 . . . . . . . . . . . . . 14 (𝑥𝑉𝑥 ∈ ((mCN‘𝑇) ∪ 𝑉))
4039ad2antlr 726 . . . . . . . . . . . . 13 ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑥𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → 𝑥 ∈ ((mCN‘𝑇) ∪ 𝑉))
4140s1cld 14550 . . . . . . . . . . . 12 ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑥𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → ⟨“𝑥”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉))
4232, 1, 2mrexval 34481 . . . . . . . . . . . . 13 (𝑇 ∈ V → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
4342ad3antrrr 729 . . . . . . . . . . . 12 ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑥𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
4441, 43eleqtrrd 2837 . . . . . . . . . . 11 ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑥𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → ⟨“𝑥”⟩ ∈ 𝑅)
4538, 44ifclda 4563 . . . . . . . . . 10 (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑥𝑉) → if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩) ∈ 𝑅)
4645fmpttd 7112 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩)):𝑉𝑅)
47 ssid 4004 . . . . . . . . 9 𝑉𝑉
4832, 1, 2, 3, 33mrsubfval 34488 . . . . . . . . 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 587 . . . . . . . 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 2783 . . . . . . 7 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑆𝑓) = (𝑆‘(𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))))
515adantr 482 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → 𝑆 Fn (𝑅pm 𝑉))
52 mapsspm 8867 . . . . . . . . 9 (𝑅m 𝑉) ⊆ (𝑅pm 𝑉)
5352a1i 11 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑅m 𝑉) ⊆ (𝑅pm 𝑉))
542fvexi 6903 . . . . . . . . . 10 𝑅 ∈ V
551fvexi 6903 . . . . . . . . . 10 𝑉 ∈ V
5654, 55elmap 8862 . . . . . . . . 9 ((𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩)) ∈ (𝑅m 𝑉) ↔ (𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩)):𝑉𝑅)
5746, 56sylibr 233 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩)) ∈ (𝑅m 𝑉))
58 fnfvima 7232 . . . . . . . 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 1372 . . . . . . 7 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑆‘(𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))) ∈ (𝑆 “ (𝑅m 𝑉)))
6050, 59eqeltrd 2834 . . . . . 6 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑆𝑓) ∈ (𝑆 “ (𝑅m 𝑉)))
6160ralrimiva 3147 . . . . 5 (𝑇 ∈ V → ∀𝑓 ∈ (𝑅pm 𝑉)(𝑆𝑓) ∈ (𝑆 “ (𝑅m 𝑉)))
62 ffnfv 7115 . . . . 5 (𝑆:(𝑅pm 𝑉)⟶(𝑆 “ (𝑅m 𝑉)) ↔ (𝑆 Fn (𝑅pm 𝑉) ∧ ∀𝑓 ∈ (𝑅pm 𝑉)(𝑆𝑓) ∈ (𝑆 “ (𝑅m 𝑉))))
635, 61, 62sylanbrc 584 . . . 4 (𝑇 ∈ V → 𝑆:(𝑅pm 𝑉)⟶(𝑆 “ (𝑅m 𝑉)))
6463frnd 6723 . . 3 (𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅m 𝑉)))
653rnfvprc 6883 . . . 4 𝑇 ∈ V → ran 𝑆 = ∅)
66 0ss 4396 . . . 4 ∅ ⊆ (𝑆 “ (𝑅m 𝑉))
6765, 66eqsstrdi 4036 . . 3 𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅m 𝑉)))
6864, 67pm2.61i 182 . 2 ran 𝑆 ⊆ (𝑆 “ (𝑅m 𝑉))
69 imassrn 6069 . 2 (𝑆 “ (𝑅m 𝑉)) ⊆ ran 𝑆
7068, 69eqssi 3998 1 ran 𝑆 = (𝑆 “ (𝑅m 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  cun 3946  wss 3948  c0 4322  ifcif 4528  cmpt 5231  dom cdm 5676  ran crn 5677  cima 5679  ccom 5680   Fn wfn 6536  wf 6537  cfv 6541  (class class class)co 7406  m cmap 8817  pm cpm 8818  Word cword 14461  ⟨“cs1 14542   Σg cgsu 17383  freeMndcfrmd 18725  mCNcmcn 34440  mVRcmvar 34441  mRExcmrex 34446  mRSubstcmrsub 34450
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-er 8700  df-map 8819  df-pm 8820  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-card 9931  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-n0 12470  df-z 12556  df-uz 12820  df-fz 13482  df-fzo 13625  df-seq 13964  df-hash 14288  df-word 14462  df-concat 14518  df-s1 14543  df-struct 17077  df-sets 17094  df-slot 17112  df-ndx 17124  df-base 17142  df-ress 17171  df-plusg 17207  df-0g 17384  df-gsum 17385  df-mgm 18558  df-sgrp 18607  df-mnd 18623  df-submnd 18669  df-frmd 18727  df-mrex 34466  df-mrsub 34470
This theorem is referenced by:  mrsubff1o  34495  mrsub0  34496  mrsubccat  34498  mrsubcn  34499  msubrn  34509
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