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 32657
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 32656 . . . . . 6 (𝑇 ∈ V → 𝑆:(𝑅pm 𝑉)⟶(𝑅m 𝑅))
54ffnd 6508 . . . . 5 (𝑇 ∈ V → 𝑆 Fn (𝑅pm 𝑉))
6 eleq1w 2892 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑣 → (𝑥 ∈ dom 𝑓𝑣 ∈ dom 𝑓))
7 fveq2 6663 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑣 → (𝑓𝑥) = (𝑓𝑣))
8 s1eq 13942 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑣 → ⟨“𝑥”⟩ = ⟨“𝑣”⟩)
96, 7, 8ifbieq12d 4490 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑣 → if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩) = if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩))
10 eqid 2818 . . . . . . . . . . . . . . . . 17 (𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩)) = (𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))
11 fvex 6676 . . . . . . . . . . . . . . . . . 18 (𝑓𝑣) ∈ V
12 s1cli 13947 . . . . . . . . . . . . . . . . . . 19 ⟨“𝑣”⟩ ∈ Word V
1312elexi 3511 . . . . . . . . . . . . . . . . . 18 ⟨“𝑣”⟩ ∈ V
1411, 13ifex 4511 . . . . . . . . . . . . . . . . 17 if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩) ∈ V
159, 10, 14fvmpt 6761 . . . . . . . . . . . . . . . 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 4493 . . . . . . . . . . . . . 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 4514 . . . . . . . . . . . . . 14 if((𝑣𝑉𝑣 ∈ dom 𝑓), (𝑓𝑣), ⟨“𝑣”⟩) = if(𝑣𝑉, if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩), ⟨“𝑣”⟩)
1917, 18syl6eqr 2871 . . . . . . . . . . . . 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 8414 . . . . . . . . . . . . . . . . . . 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 3963 . . . . . . . . . . . . . . . 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 4485 . . . . . . . . . . . . 13 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → if((𝑣𝑉𝑣 ∈ dom 𝑓), (𝑓𝑣), ⟨“𝑣”⟩) = if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩))
2719, 26eqtr2d 2854 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩) = if(𝑣𝑉, ((𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))‘𝑣), ⟨“𝑣”⟩))
2827mpteq2dv 5153 . . . . . . . . . . 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 5725 . . . . . . . . . 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 7161 . . . . . . . . 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 5153 . . . . . . . 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 2818 . . . . . . . . . 10 (mCN‘𝑇) = (mCN‘𝑇)
33 eqid 2818 . . . . . . . . . 10 (freeMnd‘((mCN‘𝑇) ∪ 𝑉)) = (freeMnd‘((mCN‘𝑇) ∪ 𝑉))
3432, 1, 2, 3, 33mrsubfval 32652 . . . . . . . . 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 6843 . . . . . . . . . . 11 ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑥𝑉) ∧ 𝑥 ∈ dom 𝑓) → (𝑓𝑥) ∈ 𝑅)
39 elun2 4150 . . . . . . . . . . . . . 14 (𝑥𝑉𝑥 ∈ ((mCN‘𝑇) ∪ 𝑉))
4039ad2antlr 723 . . . . . . . . . . . . 13 ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑥𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → 𝑥 ∈ ((mCN‘𝑇) ∪ 𝑉))
4140s1cld 13945 . . . . . . . . . . . 12 ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑥𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → ⟨“𝑥”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉))
4232, 1, 2mrexval 32645 . . . . . . . . . . . . 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 2913 . . . . . . . . . . 11 ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑥𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → ⟨“𝑥”⟩ ∈ 𝑅)
4538, 44ifclda 4497 . . . . . . . . . 10 (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑥𝑉) → if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩) ∈ 𝑅)
4645fmpttd 6871 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩)):𝑉𝑅)
47 ssid 3986 . . . . . . . . 9 𝑉𝑉
4832, 1, 2, 3, 33mrsubfval 32652 . . . . . . . . 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 2863 . . . . . . 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 8429 . . . . . . . . 9 (𝑅m 𝑉) ⊆ (𝑅pm 𝑉)
5352a1i 11 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑅m 𝑉) ⊆ (𝑅pm 𝑉))
542fvexi 6677 . . . . . . . . . 10 𝑅 ∈ V
551fvexi 6677 . . . . . . . . . 10 𝑉 ∈ V
5654, 55elmap 8424 . . . . . . . . 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 6986 . . . . . . . 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 1363 . . . . . . 7 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑆‘(𝑥𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓𝑥), ⟨“𝑥”⟩))) ∈ (𝑆 “ (𝑅m 𝑉)))
6050, 59eqeltrd 2910 . . . . . 6 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑆𝑓) ∈ (𝑆 “ (𝑅m 𝑉)))
6160ralrimiva 3179 . . . . 5 (𝑇 ∈ V → ∀𝑓 ∈ (𝑅pm 𝑉)(𝑆𝑓) ∈ (𝑆 “ (𝑅m 𝑉)))
62 ffnfv 6874 . . . . 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 6514 . . 3 (𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅m 𝑉)))
653rnfvprc 6657 . . . 4 𝑇 ∈ V → ran 𝑆 = ∅)
66 0ss 4347 . . . 4 ∅ ⊆ (𝑆 “ (𝑅m 𝑉))
6765, 66eqsstrdi 4018 . . 3 𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅m 𝑉)))
6864, 67pm2.61i 183 . 2 ran 𝑆 ⊆ (𝑆 “ (𝑅m 𝑉))
69 imassrn 5933 . 2 (𝑆 “ (𝑅m 𝑉)) ⊆ ran 𝑆
7068, 69eqssi 3980 1 ran 𝑆 = (𝑆 “ (𝑅m 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cun 3931  wss 3933  c0 4288  ifcif 4463  cmpt 5137  dom cdm 5548  ran crn 5549  cima 5551  ccom 5552   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  m cmap 8395  pm cpm 8396  Word cword 13849  ⟨“cs1 13937   Σg cgsu 16702  freeMndcfrmd 18000  mCNcmcn 32604  mVRcmvar 32605  mRExcmrex 32610  mRSubstcmrsub 32614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022  df-seq 13358  df-hash 13679  df-word 13850  df-concat 13911  df-s1 13938  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-0g 16703  df-gsum 16704  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-submnd 17945  df-frmd 18002  df-mrex 32630  df-mrsub 32634
This theorem is referenced by:  mrsubff1o  32659  mrsub0  32660  mrsubccat  32662  mrsubcn  32663  msubrn  32673
  Copyright terms: Public domain W3C validator