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Theorem elmsubrn 31724
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 2752 . . . . . 6 (mVR‘𝑇) = (mVR‘𝑇)
2 eqid 2752 . . . . . 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 31719 . . . . 5 (𝑇 ∈ V → 𝑆 = (𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑔)‘(2nd𝑒))⟩)))
71, 2, 5mrsubff 31708 . . . . . . . 8 (𝑇 ∈ V → 𝑂:((mREx‘𝑇) ↑pm (mVR‘𝑇))⟶((mREx‘𝑇) ↑𝑚 (mREx‘𝑇)))
8 ffn 6198 . . . . . . . 8 (𝑂:((mREx‘𝑇) ↑pm (mVR‘𝑇))⟶((mREx‘𝑇) ↑𝑚 (mREx‘𝑇)) → 𝑂 Fn ((mREx‘𝑇) ↑pm (mVR‘𝑇)))
97, 8syl 17 . . . . . . 7 (𝑇 ∈ V → 𝑂 Fn ((mREx‘𝑇) ↑pm (mVR‘𝑇)))
10 fnfvelrn 6511 . . . . . . 7 ((𝑂 Fn ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ∧ 𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇))) → (𝑂𝑔) ∈ ran 𝑂)
119, 10sylan 489 . . . . . 6 ((𝑇 ∈ V ∧ 𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇))) → (𝑂𝑔) ∈ ran 𝑂)
127feqmptd 6403 . . . . . 6 (𝑇 ∈ V → 𝑂 = (𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ↦ (𝑂𝑔)))
13 eqidd 2753 . . . . . 6 (𝑇 ∈ V → (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) = (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
14 fveq1 6343 . . . . . . . 8 (𝑓 = (𝑂𝑔) → (𝑓‘(2nd𝑒)) = ((𝑂𝑔)‘(2nd𝑒)))
1514opeq2d 4552 . . . . . . 7 (𝑓 = (𝑂𝑔) → ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩ = ⟨(1st𝑒), ((𝑂𝑔)‘(2nd𝑒))⟩)
1615mpteq2dv 4889 . . . . . 6 (𝑓 = (𝑂𝑔) → (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑔)‘(2nd𝑒))⟩))
1711, 12, 13, 16fmptco 6551 . . . . 5 (𝑇 ∈ V → ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ∘ 𝑂) = (𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑔)‘(2nd𝑒))⟩)))
186, 17eqtr4d 2789 . . . 4 (𝑇 ∈ V → 𝑆 = ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ∘ 𝑂))
1918rneqd 5500 . . 3 (𝑇 ∈ V → ran 𝑆 = ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ∘ 𝑂))
20 rnco 5794 . . . 4 ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ∘ 𝑂) = ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ↾ ran 𝑂)
21 ssid 3757 . . . . . 6 ran 𝑂 ⊆ ran 𝑂
22 resmpt 5599 . . . . . 6 (ran 𝑂 ⊆ ran 𝑂 → ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ↾ ran 𝑂) = (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
2321, 22ax-mp 5 . . . . 5 ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ↾ ran 𝑂) = (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
2423rneqi 5499 . . . 4 ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ↾ ran 𝑂) = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
2520, 24eqtri 2774 . . 3 ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ∘ 𝑂) = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
2619, 25syl6eq 2802 . 2 (𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
27 mpt0 6174 . . . . 5 (𝑓 ∈ ∅ ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) = ∅
2827eqcomi 2761 . . . 4 ∅ = (𝑓 ∈ ∅ ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
29 fvprc 6338 . . . . 5 𝑇 ∈ V → (mSubst‘𝑇) = ∅)
303, 29syl5eq 2798 . . . 4 𝑇 ∈ V → 𝑆 = ∅)
31 fvprc 6338 . . . . . . . 8 𝑇 ∈ V → (mRSubst‘𝑇) = ∅)
325, 31syl5eq 2798 . . . . . . 7 𝑇 ∈ V → 𝑂 = ∅)
3332rneqd 5500 . . . . . 6 𝑇 ∈ V → ran 𝑂 = ran ∅)
34 rn0 5524 . . . . . 6 ran ∅ = ∅
3533, 34syl6eq 2802 . . . . 5 𝑇 ∈ V → ran 𝑂 = ∅)
3635mpteq1d 4882 . . . 4 𝑇 ∈ V → (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) = (𝑓 ∈ ∅ ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
3728, 30, 363eqtr4a 2812 . . 3 𝑇 ∈ V → 𝑆 = (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
3837rneqd 5500 . 2 𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
3926, 38pm2.61i 176 1 ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1624  wcel 2131  Vcvv 3332  wss 3707  c0 4050  cop 4319  cmpt 4873  ran crn 5259  cres 5260  ccom 5262   Fn wfn 6036  wf 6037  cfv 6041  (class class class)co 6805  1st c1st 7323  2nd c2nd 7324  𝑚 cmap 8015  pm cpm 8016  mVRcmvar 31657  mRExcmrex 31662  mExcmex 31663  mRSubstcmrsub 31666  mSubstcmsub 31667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-er 7903  df-map 8017  df-pm 8018  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-card 8947  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-n0 11477  df-z 11562  df-uz 11872  df-fz 12512  df-fzo 12652  df-seq 12988  df-hash 13304  df-word 13477  df-concat 13479  df-s1 13480  df-struct 16053  df-ndx 16054  df-slot 16055  df-base 16057  df-sets 16058  df-ress 16059  df-plusg 16148  df-0g 16296  df-gsum 16297  df-mgm 17435  df-sgrp 17477  df-mnd 17488  df-submnd 17529  df-frmd 17579  df-mrex 31682  df-mrsub 31686  df-msub 31687
This theorem is referenced by:  msubco  31727  msubvrs  31756
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