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Theorem mrsubff 32759
 Description: A substitution is a function from 𝑅 to 𝑅. (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
mrsubff (𝑇𝑊𝑆:(𝑅pm 𝑉)⟶(𝑅m 𝑅))

Proof of Theorem mrsubff
Dummy variables 𝑒 𝑓 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6682 . . . . . . . . 9 (mCN‘𝑇) ∈ V
2 mrsubvr.v . . . . . . . . . 10 𝑉 = (mVR‘𝑇)
32fvexi 6683 . . . . . . . . 9 𝑉 ∈ V
41, 3unex 7468 . . . . . . . 8 ((mCN‘𝑇) ∪ 𝑉) ∈ V
5 eqid 2821 . . . . . . . . 9 (freeMnd‘((mCN‘𝑇) ∪ 𝑉)) = (freeMnd‘((mCN‘𝑇) ∪ 𝑉))
65frmdmnd 18023 . . . . . . . 8 (((mCN‘𝑇) ∪ 𝑉) ∈ V → (freeMnd‘((mCN‘𝑇) ∪ 𝑉)) ∈ Mnd)
74, 6mp1i 13 . . . . . . 7 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → (freeMnd‘((mCN‘𝑇) ∪ 𝑉)) ∈ Mnd)
8 simpr 487 . . . . . . . . 9 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → 𝑒𝑅)
9 eqid 2821 . . . . . . . . . . 11 (mCN‘𝑇) = (mCN‘𝑇)
10 mrsubvr.r . . . . . . . . . . 11 𝑅 = (mREx‘𝑇)
119, 2, 10mrexval 32748 . . . . . . . . . 10 (𝑇𝑊𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
1211ad2antrr 724 . . . . . . . . 9 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
138, 12eleqtrd 2915 . . . . . . . 8 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → 𝑒 ∈ Word ((mCN‘𝑇) ∪ 𝑉))
14 elpmi 8424 . . . . . . . . . . . . . 14 (𝑓 ∈ (𝑅pm 𝑉) → (𝑓:dom 𝑓𝑅 ∧ dom 𝑓𝑉))
1514simpld 497 . . . . . . . . . . . . 13 (𝑓 ∈ (𝑅pm 𝑉) → 𝑓:dom 𝑓𝑅)
1615ad3antlr 729 . . . . . . . . . . . 12 ((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) → 𝑓:dom 𝑓𝑅)
1716ffvelrnda 6850 . . . . . . . . . . 11 (((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ 𝑣 ∈ dom 𝑓) → (𝑓𝑣) ∈ 𝑅)
1812ad2antrr 724 . . . . . . . . . . 11 (((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ 𝑣 ∈ dom 𝑓) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
1917, 18eleqtrd 2915 . . . . . . . . . 10 (((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ 𝑣 ∈ dom 𝑓) → (𝑓𝑣) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
20 simplr 767 . . . . . . . . . . 11 (((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ ¬ 𝑣 ∈ dom 𝑓) → 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉))
2120s1cld 13956 . . . . . . . . . 10 (((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ ¬ 𝑣 ∈ dom 𝑓) → ⟨“𝑣”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉))
2219, 21ifclda 4500 . . . . . . . . 9 ((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) → if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
2322fmpttd 6878 . . . . . . . 8 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → (𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)):((mCN‘𝑇) ∪ 𝑉)⟶Word ((mCN‘𝑇) ∪ 𝑉))
24 wrdco 14192 . . . . . . . 8 ((𝑒 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ (𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)):((mCN‘𝑇) ∪ 𝑉)⟶Word ((mCN‘𝑇) ∪ 𝑉)) → ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒) ∈ Word Word ((mCN‘𝑇) ∪ 𝑉))
2513, 23, 24syl2anc 586 . . . . . . 7 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒) ∈ Word Word ((mCN‘𝑇) ∪ 𝑉))
26 eqid 2821 . . . . . . . . . . 11 (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉))) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉)))
275, 26frmdbas 18016 . . . . . . . . . 10 (((mCN‘𝑇) ∪ 𝑉) ∈ V → (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉))) = Word ((mCN‘𝑇) ∪ 𝑉))
284, 27ax-mp 5 . . . . . . . . 9 (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉))) = Word ((mCN‘𝑇) ∪ 𝑉)
2928eqcomi 2830 . . . . . . . 8 Word ((mCN‘𝑇) ∪ 𝑉) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉)))
3029gsumwcl 18002 . . . . . . 7 (((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) ∈ Mnd ∧ ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒) ∈ Word Word ((mCN‘𝑇) ∪ 𝑉)) → ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
317, 25, 30syl2anc 586 . . . . . 6 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
3231, 12eleqtrrd 2916 . . . . 5 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)) ∈ 𝑅)
3332fmpttd 6878 . . . 4 ((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) → (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))):𝑅𝑅)
3410fvexi 6683 . . . . 5 𝑅 ∈ V
3534, 34elmap 8434 . . . 4 ((𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) ∈ (𝑅m 𝑅) ↔ (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))):𝑅𝑅)
3633, 35sylibr 236 . . 3 ((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) → (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) ∈ (𝑅m 𝑅))
3736fmpttd 6878 . 2 (𝑇𝑊 → (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))):(𝑅pm 𝑉)⟶(𝑅m 𝑅))
38 mrsubvr.s . . . 4 𝑆 = (mRSubst‘𝑇)
399, 2, 10, 38, 5mrsubffval 32754 . . 3 (𝑇𝑊𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
4039feq1d 6498 . 2 (𝑇𝑊 → (𝑆:(𝑅pm 𝑉)⟶(𝑅m 𝑅) ↔ (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))):(𝑅pm 𝑉)⟶(𝑅m 𝑅)))
4137, 40mpbird 259 1 (𝑇𝑊𝑆:(𝑅pm 𝑉)⟶(𝑅m 𝑅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Vcvv 3494   ∪ cun 3933   ⊆ wss 3935  ifcif 4466   ↦ cmpt 5145  dom cdm 5554   ∘ ccom 5558  ⟶wf 6350  ‘cfv 6354  (class class class)co 7155   ↑m cmap 8405   ↑pm cpm 8406  Word cword 13860  ⟨“cs1 13948  Basecbs 16482   Σg cgsu 16713  Mndcmnd 17910  freeMndcfrmd 18011  mCNcmcn 32707  mVRcmvar 32708  mRExcmrex 32713  mRSubstcmrsub 32717 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  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 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-er 8288  df-map 8407  df-pm 8408  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-card 9367  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-nn 11638  df-2 11699  df-n0 11897  df-z 11981  df-uz 12243  df-fz 12892  df-fzo 13033  df-seq 13369  df-hash 13690  df-word 13861  df-concat 13922  df-s1 13949  df-struct 16484  df-ndx 16485  df-slot 16486  df-base 16488  df-sets 16489  df-ress 16490  df-plusg 16577  df-0g 16714  df-gsum 16715  df-mgm 17851  df-sgrp 17900  df-mnd 17911  df-submnd 17956  df-frmd 18013  df-mrex 32733  df-mrsub 32737 This theorem is referenced by:  mrsubrn  32760  mrsubff1  32761  mrsub0  32763  mrsubf  32764  mrsubccat  32765  mrsubcn  32766  elmrsubrn  32767  elmsubrn  32775  msubrn  32776  msubff  32777  msubff1  32803
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