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

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
  Copyright terms: Public domain W3C validator