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Theorem mrsubff 30497
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 6098 . . . . . . . . 9 (mCN‘𝑇) ∈ V
2 mrsubvr.v . . . . . . . . . 10 𝑉 = (mVR‘𝑇)
3 fvex 6098 . . . . . . . . . 10 (mVR‘𝑇) ∈ V
42, 3eqeltri 2683 . . . . . . . . 9 𝑉 ∈ V
51, 4unex 6832 . . . . . . . 8 ((mCN‘𝑇) ∪ 𝑉) ∈ V
6 eqid 2609 . . . . . . . . 9 (freeMnd‘((mCN‘𝑇) ∪ 𝑉)) = (freeMnd‘((mCN‘𝑇) ∪ 𝑉))
76frmdmnd 17168 . . . . . . . 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 475 . . . . . . . . 9 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → 𝑒𝑅)
10 eqid 2609 . . . . . . . . . . 11 (mCN‘𝑇) = (mCN‘𝑇)
11 mrsubvr.r . . . . . . . . . . 11 𝑅 = (mREx‘𝑇)
1210, 2, 11mrexval 30486 . . . . . . . . . 10 (𝑇𝑊𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
1312ad2antrr 757 . . . . . . . . 9 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
149, 13eleqtrd 2689 . . . . . . . 8 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → 𝑒 ∈ Word ((mCN‘𝑇) ∪ 𝑉))
15 elpmi 7740 . . . . . . . . . . . . . 14 (𝑓 ∈ (𝑅pm 𝑉) → (𝑓:dom 𝑓𝑅 ∧ dom 𝑓𝑉))
1615simpld 473 . . . . . . . . . . . . 13 (𝑓 ∈ (𝑅pm 𝑉) → 𝑓:dom 𝑓𝑅)
1716ad3antlr 762 . . . . . . . . . . . 12 ((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) → 𝑓:dom 𝑓𝑅)
1817ffvelrnda 6252 . . . . . . . . . . 11 (((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ 𝑣 ∈ dom 𝑓) → (𝑓𝑣) ∈ 𝑅)
1913ad2antrr 757 . . . . . . . . . . 11 (((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ 𝑣 ∈ dom 𝑓) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
2018, 19eleqtrd 2689 . . . . . . . . . 10 (((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ 𝑣 ∈ dom 𝑓) → (𝑓𝑣) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
21 simplr 787 . . . . . . . . . . 11 (((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ ¬ 𝑣 ∈ dom 𝑓) → 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉))
2221s1cld 13185 . . . . . . . . . 10 (((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ ¬ 𝑣 ∈ dom 𝑓) → ⟨“𝑣”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉))
2320, 22ifclda 4069 . . . . . . . . 9 ((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) → if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
24 eqid 2609 . . . . . . . . 9 (𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) = (𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩))
2523, 24fmptd 6277 . . . . . . . 8 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → (𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)):((mCN‘𝑇) ∪ 𝑉)⟶Word ((mCN‘𝑇) ∪ 𝑉))
26 wrdco 13377 . . . . . . . 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 690 . . . . . . 7 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒) ∈ Word Word ((mCN‘𝑇) ∪ 𝑉))
28 eqid 2609 . . . . . . . . . . 11 (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉))) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉)))
296, 28frmdbas 17161 . . . . . . . . . 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 2618 . . . . . . . 8 Word ((mCN‘𝑇) ∪ 𝑉) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉)))
3231gsumwcl 17149 . . . . . . 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 690 . . . . . 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 2690 . . . . 5 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)) ∈ 𝑅)
35 eqid 2609 . . . . 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 6277 . . . 4 ((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) → (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))):𝑅𝑅)
37 fvex 6098 . . . . . 6 (mREx‘𝑇) ∈ V
3811, 37eqeltri 2683 . . . . 5 𝑅 ∈ V
3938, 38elmap 7750 . . . 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 222 . . 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 2609 . . 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 6277 . 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 30492 . . 3 (𝑇𝑊𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
4544feq1d 5929 . 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 245 1 (𝑇𝑊𝑆:(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3172  cun 3537  wss 3539  ifcif 4035  cmpt 4637  dom cdm 5028  ccom 5032  wf 5786  cfv 5790  (class class class)co 6527  𝑚 cmap 7722  pm cpm 7723  Word cword 13095  ⟨“cs1 13098  Basecbs 15644   Σg cgsu 15873  Mndcmnd 17066  freeMndcfrmd 17156  mCNcmcn 30445  mVRcmvar 30446  mRExcmrex 30451  mRSubstcmrsub 30455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-oadd 7429  df-er 7607  df-map 7724  df-pm 7725  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-card 8626  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-nn 10871  df-2 10929  df-n0 11143  df-z 11214  df-uz 11523  df-fz 12156  df-fzo 12293  df-seq 12622  df-hash 12938  df-word 13103  df-concat 13105  df-s1 13106  df-struct 15646  df-ndx 15647  df-slot 15648  df-base 15649  df-sets 15650  df-ress 15651  df-plusg 15730  df-0g 15874  df-gsum 15875  df-mgm 17014  df-sgrp 17056  df-mnd 17067  df-submnd 17108  df-frmd 17158  df-mrex 30471  df-mrsub 30475
This theorem is referenced by:  mrsubrn  30498  mrsubff1  30499  mrsub0  30501  mrsubf  30502  mrsubccat  30503  mrsubcn  30504  elmrsubrn  30505  elmsubrn  30513  msubrn  30514  msubff  30515  msubff1  30541
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