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Theorem mrsubff 31535
 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 6239 . . . . . . . . 9 (mCN‘𝑇) ∈ V
2 mrsubvr.v . . . . . . . . . 10 𝑉 = (mVR‘𝑇)
3 fvex 6239 . . . . . . . . . 10 (mVR‘𝑇) ∈ V
42, 3eqeltri 2726 . . . . . . . . 9 𝑉 ∈ V
51, 4unex 6998 . . . . . . . 8 ((mCN‘𝑇) ∪ 𝑉) ∈ V
6 eqid 2651 . . . . . . . . 9 (freeMnd‘((mCN‘𝑇) ∪ 𝑉)) = (freeMnd‘((mCN‘𝑇) ∪ 𝑉))
76frmdmnd 17443 . . . . . . . 8 (((mCN‘𝑇) ∪ 𝑉) ∈ V → (freeMnd‘((mCN‘𝑇) ∪ 𝑉)) ∈ Mnd)
85, 7mp1i 13 . . . . . . 7 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → (freeMnd‘((mCN‘𝑇) ∪ 𝑉)) ∈ Mnd)
9 simpr 476 . . . . . . . . 9 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → 𝑒𝑅)
10 eqid 2651 . . . . . . . . . . 11 (mCN‘𝑇) = (mCN‘𝑇)
11 mrsubvr.r . . . . . . . . . . 11 𝑅 = (mREx‘𝑇)
1210, 2, 11mrexval 31524 . . . . . . . . . 10 (𝑇𝑊𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
1312ad2antrr 762 . . . . . . . . 9 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
149, 13eleqtrd 2732 . . . . . . . 8 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → 𝑒 ∈ Word ((mCN‘𝑇) ∪ 𝑉))
15 elpmi 7918 . . . . . . . . . . . . . 14 (𝑓 ∈ (𝑅pm 𝑉) → (𝑓:dom 𝑓𝑅 ∧ dom 𝑓𝑉))
1615simpld 474 . . . . . . . . . . . . 13 (𝑓 ∈ (𝑅pm 𝑉) → 𝑓:dom 𝑓𝑅)
1716ad3antlr 767 . . . . . . . . . . . 12 ((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) → 𝑓:dom 𝑓𝑅)
1817ffvelrnda 6399 . . . . . . . . . . 11 (((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ 𝑣 ∈ dom 𝑓) → (𝑓𝑣) ∈ 𝑅)
1913ad2antrr 762 . . . . . . . . . . 11 (((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ 𝑣 ∈ dom 𝑓) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
2018, 19eleqtrd 2732 . . . . . . . . . 10 (((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ 𝑣 ∈ dom 𝑓) → (𝑓𝑣) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
21 simplr 807 . . . . . . . . . . 11 (((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ ¬ 𝑣 ∈ dom 𝑓) → 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉))
2221s1cld 13419 . . . . . . . . . 10 (((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ ¬ 𝑣 ∈ dom 𝑓) → ⟨“𝑣”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉))
2320, 22ifclda 4153 . . . . . . . . 9 ((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) → if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
24 eqid 2651 . . . . . . . . 9 (𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) = (𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩))
2523, 24fmptd 6425 . . . . . . . 8 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → (𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)):((mCN‘𝑇) ∪ 𝑉)⟶Word ((mCN‘𝑇) ∪ 𝑉))
26 wrdco 13623 . . . . . . . 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))
2714, 25, 26syl2anc 694 . . . . . . 7 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒) ∈ Word Word ((mCN‘𝑇) ∪ 𝑉))
28 eqid 2651 . . . . . . . . . . 11 (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉))) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉)))
296, 28frmdbas 17436 . . . . . . . . . 10 (((mCN‘𝑇) ∪ 𝑉) ∈ V → (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉))) = Word ((mCN‘𝑇) ∪ 𝑉))
305, 29ax-mp 5 . . . . . . . . 9 (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉))) = Word ((mCN‘𝑇) ∪ 𝑉)
3130eqcomi 2660 . . . . . . . 8 Word ((mCN‘𝑇) ∪ 𝑉) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉)))
3231gsumwcl 17424 . . . . . . 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))
338, 27, 32syl2anc 694 . . . . . 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))
3433, 13eleqtrrd 2733 . . . . 5 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)) ∈ 𝑅)
35 eqid 2651 . . . . 5 (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(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))
3634, 35fmptd 6425 . . . 4 ((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) → (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))):𝑅𝑅)
37 fvex 6239 . . . . . 6 (mREx‘𝑇) ∈ V
3811, 37eqeltri 2726 . . . . 5 𝑅 ∈ V
3938, 38elmap 7928 . . . 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)
4036, 39sylibr 224 . . 3 ((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) → (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) ∈ (𝑅𝑚 𝑅))
41 eqid 2651 . . 3 (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))) = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
4240, 41fmptd 6425 . 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))
43 mrsubvr.s . . . 4 𝑆 = (mRSubst‘𝑇)
4410, 2, 11, 43, 6mrsubffval 31530 . . 3 (𝑇𝑊𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
4544feq1d 6068 . 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)))
4642, 45mpbird 247 1 (𝑇𝑊𝑆:(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∪ cun 3605   ⊆ wss 3607  ifcif 4119   ↦ cmpt 4762  dom cdm 5143   ∘ ccom 5147  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ↑𝑚 cmap 7899   ↑pm cpm 7900  Word cword 13323  ⟨“cs1 13326  Basecbs 15904   Σg cgsu 16148  Mndcmnd 17341  freeMndcfrmd 17431  mCNcmcn 31483  mVRcmvar 31484  mRExcmrex 31489  mRSubstcmrsub 31493 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-word 13331  df-concat 13333  df-s1 13334  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-0g 16149  df-gsum 16150  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-frmd 17433  df-mrex 31509  df-mrsub 31513 This theorem is referenced by:  mrsubrn  31536  mrsubff1  31537  mrsub0  31539  mrsubf  31540  mrsubccat  31541  mrsubcn  31542  elmrsubrn  31543  elmsubrn  31551  msubrn  31552  msubff  31553  msubff1  31579
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