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Theorem elmsubrn 30679
 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 2610 . . . . . 6 (mVR‘𝑇) = (mVR‘𝑇)
2 eqid 2610 . . . . . 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 30674 . . . . 5 (𝑇 ∈ V → 𝑆 = (𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑔)‘(2nd𝑒))⟩)))
71, 2, 5mrsubff 30663 . . . . . . . 8 (𝑇 ∈ V → 𝑂:((mREx‘𝑇) ↑pm (mVR‘𝑇))⟶((mREx‘𝑇) ↑𝑚 (mREx‘𝑇)))
8 ffn 5958 . . . . . . . 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 6264 . . . . . . 7 ((𝑂 Fn ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ∧ 𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇))) → (𝑂𝑔) ∈ ran 𝑂)
119, 10sylan 487 . . . . . 6 ((𝑇 ∈ V ∧ 𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇))) → (𝑂𝑔) ∈ ran 𝑂)
127feqmptd 6159 . . . . . 6 (𝑇 ∈ V → 𝑂 = (𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ↦ (𝑂𝑔)))
13 eqidd 2611 . . . . . 6 (𝑇 ∈ V → (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) = (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
14 fveq1 6102 . . . . . . . 8 (𝑓 = (𝑂𝑔) → (𝑓‘(2nd𝑒)) = ((𝑂𝑔)‘(2nd𝑒)))
1514opeq2d 4347 . . . . . . 7 (𝑓 = (𝑂𝑔) → ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩ = ⟨(1st𝑒), ((𝑂𝑔)‘(2nd𝑒))⟩)
1615mpteq2dv 4673 . . . . . 6 (𝑓 = (𝑂𝑔) → (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑔)‘(2nd𝑒))⟩))
1711, 12, 13, 16fmptco 6303 . . . . 5 (𝑇 ∈ V → ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ∘ 𝑂) = (𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑔)‘(2nd𝑒))⟩)))
186, 17eqtr4d 2647 . . . 4 (𝑇 ∈ V → 𝑆 = ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ∘ 𝑂))
1918rneqd 5274 . . 3 (𝑇 ∈ V → ran 𝑆 = ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ∘ 𝑂))
20 rnco 5558 . . . 4 ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ∘ 𝑂) = ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ↾ ran 𝑂)
21 ssid 3587 . . . . . 6 ran 𝑂 ⊆ ran 𝑂
22 resmpt 5369 . . . . . 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 5273 . . . 4 ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ↾ ran 𝑂) = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
2520, 24eqtri 2632 . . 3 ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) ∘ 𝑂) = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
2619, 25syl6eq 2660 . 2 (𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
27 mpt0 5934 . . . . 5 (𝑓 ∈ ∅ ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) = ∅
2827eqcomi 2619 . . . 4 ∅ = (𝑓 ∈ ∅ ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
29 fvprc 6097 . . . . 5 𝑇 ∈ V → (mSubst‘𝑇) = ∅)
303, 29syl5eq 2656 . . . 4 𝑇 ∈ V → 𝑆 = ∅)
31 fvprc 6097 . . . . . . . 8 𝑇 ∈ V → (mRSubst‘𝑇) = ∅)
325, 31syl5eq 2656 . . . . . . 7 𝑇 ∈ V → 𝑂 = ∅)
3332rneqd 5274 . . . . . 6 𝑇 ∈ V → ran 𝑂 = ran ∅)
34 rn0 5298 . . . . . 6 ran ∅ = ∅
3533, 34syl6eq 2660 . . . . 5 𝑇 ∈ V → ran 𝑂 = ∅)
3635mpteq1d 4666 . . . 4 𝑇 ∈ V → (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)) = (𝑓 ∈ ∅ ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
3728, 30, 363eqtr4a 2670 . . 3 𝑇 ∈ V → 𝑆 = (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
3837rneqd 5274 . 2 𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩)))
3926, 38pm2.61i 175 1 ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ⟨cop 4131   ↦ cmpt 4643  ran crn 5039   ↾ cres 5040   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058   ↑𝑚 cmap 7744   ↑pm cpm 7745  mVRcmvar 30612  mRExcmrex 30617  mExcmex 30618  mRSubstcmrsub 30621  mSubstcmsub 30622 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-gsum 15926  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-frmd 17209  df-mrex 30637  df-mrsub 30641  df-msub 30642 This theorem is referenced by:  msubco  30682  msubvrs  30711
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