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Theorem elmsubrn 32024
 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 2778 . . . . . 6 (mVR‘𝑇) = (mVR‘𝑇)
2 eqid 2778 . . . . . 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 32019 . . . . 5 (𝑇 ∈ V → 𝑆 = (𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑔)‘(2nd𝑒))⟩)))
71, 2, 5mrsubff 32008 . . . . . . . 8 (𝑇 ∈ V → 𝑂:((mREx‘𝑇) ↑pm (mVR‘𝑇))⟶((mREx‘𝑇) ↑𝑚 (mREx‘𝑇)))
87ffnd 6292 . . . . . . 7 (𝑇 ∈ V → 𝑂 Fn ((mREx‘𝑇) ↑pm (mVR‘𝑇)))
9 fnfvelrn 6620 . . . . . . 7 ((𝑂 Fn ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ∧ 𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇))) → (𝑂𝑔) ∈ ran 𝑂)
108, 9sylan 575 . . . . . 6 ((𝑇 ∈ V ∧ 𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇))) → (𝑂𝑔) ∈ ran 𝑂)
117feqmptd 6509 . . . . . 6 (𝑇 ∈ V → 𝑂 = (𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ↦ (𝑂𝑔)))
12 eqidd 2779 . . . . . 6 (𝑇 ∈ V → (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) = (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
13 fveq1 6445 . . . . . . . 8 (𝑓 = (𝑂𝑔) → (𝑓‘(2nd𝑒)) = ((𝑂𝑔)‘(2nd𝑒)))
1413opeq2d 4643 . . . . . . 7 (𝑓 = (𝑂𝑔) → ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩ = ⟨(1st𝑒), ((𝑂𝑔)‘(2nd𝑒))⟩)
1514mpteq2dv 4980 . . . . . 6 (𝑓 = (𝑂𝑔) → (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑔)‘(2nd𝑒))⟩))
1610, 11, 12, 15fmptco 6661 . . . . 5 (𝑇 ∈ V → ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ∘ 𝑂) = (𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑔)‘(2nd𝑒))⟩)))
176, 16eqtr4d 2817 . . . 4 (𝑇 ∈ V → 𝑆 = ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ∘ 𝑂))
1817rneqd 5598 . . 3 (𝑇 ∈ V → ran 𝑆 = ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ∘ 𝑂))
19 rnco 5895 . . . 4 ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ∘ 𝑂) = ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ↾ ran 𝑂)
20 ssid 3842 . . . . . 6 ran 𝑂 ⊆ ran 𝑂
21 resmpt 5699 . . . . . 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 5597 . . . 4 ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ↾ ran 𝑂) = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
2419, 23eqtri 2802 . . 3 ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ∘ 𝑂) = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
2518, 24syl6eq 2830 . 2 (𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
26 mpt0 6267 . . . . 5 (𝑓 ∈ ∅ ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) = ∅
2726eqcomi 2787 . . . 4 ∅ = (𝑓 ∈ ∅ ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
28 fvprc 6439 . . . . 5 𝑇 ∈ V → (mSubst‘𝑇) = ∅)
293, 28syl5eq 2826 . . . 4 𝑇 ∈ V → 𝑆 = ∅)
305rnfvprc 6440 . . . . 5 𝑇 ∈ V → ran 𝑂 = ∅)
3130mpteq1d 4973 . . . 4 𝑇 ∈ V → (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) = (𝑓 ∈ ∅ ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
3227, 29, 313eqtr4a 2840 . . 3 𝑇 ∈ V → 𝑆 = (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
3332rneqd 5598 . 2 𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
3425, 33pm2.61i 177 1 ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1601   ∈ wcel 2107  Vcvv 3398   ⊆ wss 3792  ∅c0 4141  ⟨cop 4404   ↦ cmpt 4965  ran crn 5356   ↾ cres 5357   ∘ ccom 5359   Fn wfn 6130  ‘cfv 6135  (class class class)co 6922  1st c1st 7443  2nd c2nd 7444   ↑𝑚 cmap 8140   ↑pm cpm 8141  mVRcmvar 31957  mRExcmrex 31962  mExcmex 31963  mRSubstcmrsub 31966  mSubstcmsub 31967 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-n0 11643  df-z 11729  df-uz 11993  df-fz 12644  df-fzo 12785  df-seq 13120  df-hash 13436  df-word 13600  df-concat 13661  df-s1 13686  df-struct 16257  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-ress 16263  df-plusg 16351  df-0g 16488  df-gsum 16489  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-submnd 17722  df-frmd 17773  df-mrex 31982  df-mrsub 31986  df-msub 31987 This theorem is referenced by:  msubco  32027  msubvrs  32056
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