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Theorem elmsubrn 32849
Description: Characterization of substitution in terms of raw substitution, without reference to the generating functions. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
elmsubrn.e 𝐸 = (mEx‘𝑇)
elmsubrn.o 𝑂 = (mRSubst‘𝑇)
elmsubrn.s 𝑆 = (mSubst‘𝑇)
Assertion
Ref Expression
elmsubrn ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
Distinct variable groups:   𝑒,𝑓,𝐸   𝑒,𝑂,𝑓   𝑇,𝑒
Allowed substitution hints:   𝑆(𝑒,𝑓)   𝑇(𝑓)

Proof of Theorem elmsubrn
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 eqid 2822 . . . . . 6 (mVR‘𝑇) = (mVR‘𝑇)
2 eqid 2822 . . . . . 6 (mREx‘𝑇) = (mREx‘𝑇)
3 elmsubrn.s . . . . . 6 𝑆 = (mSubst‘𝑇)
4 elmsubrn.e . . . . . 6 𝐸 = (mEx‘𝑇)
5 elmsubrn.o . . . . . 6 𝑂 = (mRSubst‘𝑇)
61, 2, 3, 4, 5msubffval 32844 . . . . 5 (𝑇 ∈ V → 𝑆 = (𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑔)‘(2nd𝑒))⟩)))
71, 2, 5mrsubff 32833 . . . . . . . 8 (𝑇 ∈ V → 𝑂:((mREx‘𝑇) ↑pm (mVR‘𝑇))⟶((mREx‘𝑇) ↑m (mREx‘𝑇)))
87ffnd 6495 . . . . . . 7 (𝑇 ∈ V → 𝑂 Fn ((mREx‘𝑇) ↑pm (mVR‘𝑇)))
9 fnfvelrn 6830 . . . . . . 7 ((𝑂 Fn ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ∧ 𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇))) → (𝑂𝑔) ∈ ran 𝑂)
108, 9sylan 583 . . . . . 6 ((𝑇 ∈ V ∧ 𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇))) → (𝑂𝑔) ∈ ran 𝑂)
117feqmptd 6715 . . . . . 6 (𝑇 ∈ V → 𝑂 = (𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ↦ (𝑂𝑔)))
12 eqidd 2823 . . . . . 6 (𝑇 ∈ V → (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) = (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
13 fveq1 6651 . . . . . . . 8 (𝑓 = (𝑂𝑔) → (𝑓‘(2nd𝑒)) = ((𝑂𝑔)‘(2nd𝑒)))
1413opeq2d 4785 . . . . . . 7 (𝑓 = (𝑂𝑔) → ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩ = ⟨(1st𝑒), ((𝑂𝑔)‘(2nd𝑒))⟩)
1514mpteq2dv 5138 . . . . . 6 (𝑓 = (𝑂𝑔) → (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑔)‘(2nd𝑒))⟩))
1610, 11, 12, 15fmptco 6873 . . . . 5 (𝑇 ∈ V → ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ∘ 𝑂) = (𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑔)‘(2nd𝑒))⟩)))
176, 16eqtr4d 2860 . . . 4 (𝑇 ∈ V → 𝑆 = ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ∘ 𝑂))
1817rneqd 5785 . . 3 (𝑇 ∈ V → ran 𝑆 = ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ∘ 𝑂))
19 rnco 6083 . . . 4 ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ∘ 𝑂) = ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ↾ ran 𝑂)
20 ssid 3964 . . . . . 6 ran 𝑂 ⊆ ran 𝑂
21 resmpt 5883 . . . . . 6 (ran 𝑂 ⊆ ran 𝑂 → ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ↾ ran 𝑂) = (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
2220, 21ax-mp 5 . . . . 5 ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ↾ ran 𝑂) = (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
2322rneqi 5784 . . . 4 ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ↾ ran 𝑂) = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
2419, 23eqtri 2845 . . 3 ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ∘ 𝑂) = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
2518, 24syl6eq 2873 . 2 (𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
26 mpt0 6470 . . . . 5 (𝑓 ∈ ∅ ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) = ∅
2726eqcomi 2831 . . . 4 ∅ = (𝑓 ∈ ∅ ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
28 fvprc 6645 . . . . 5 𝑇 ∈ V → (mSubst‘𝑇) = ∅)
293, 28syl5eq 2869 . . . 4 𝑇 ∈ V → 𝑆 = ∅)
305rnfvprc 6646 . . . . 5 𝑇 ∈ V → ran 𝑂 = ∅)
3130mpteq1d 5131 . . . 4 𝑇 ∈ V → (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) = (𝑓 ∈ ∅ ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
3227, 29, 313eqtr4a 2883 . . 3 𝑇 ∈ V → 𝑆 = (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
3332rneqd 5785 . 2 𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
3425, 33pm2.61i 185 1 ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1538  wcel 2114  Vcvv 3469  wss 3908  c0 4265  cop 4545  cmpt 5122  ran crn 5533  cres 5534  ccom 5536   Fn wfn 6329  cfv 6334  (class class class)co 7140  1st c1st 7673  2nd c2nd 7674  m cmap 8393  pm cpm 8394  mVRcmvar 32782  mRExcmrex 32787  mExcmex 32788  mRSubstcmrsub 32791  mSubstcmsub 32792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-card 9356  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-seq 13365  df-hash 13687  df-word 13858  df-concat 13914  df-s1 13941  df-struct 16476  df-ndx 16477  df-slot 16478  df-base 16480  df-sets 16481  df-ress 16482  df-plusg 16569  df-0g 16706  df-gsum 16707  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-submnd 17948  df-frmd 18005  df-mrex 32807  df-mrsub 32811  df-msub 32812
This theorem is referenced by:  msubco  32852  msubvrs  32881
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