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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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