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Theorem elmsubrn 35171
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 = (mExβ€˜π‘‡)
elmsubrn.o 𝑂 = (mRSubstβ€˜π‘‡)
elmsubrn.s 𝑆 = (mSubstβ€˜π‘‡)
Assertion
Ref Expression
elmsubrn ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩))
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 2728 . . . . . 6 (mVRβ€˜π‘‡) = (mVRβ€˜π‘‡)
2 eqid 2728 . . . . . 6 (mRExβ€˜π‘‡) = (mRExβ€˜π‘‡)
3 elmsubrn.s . . . . . 6 𝑆 = (mSubstβ€˜π‘‡)
4 elmsubrn.e . . . . . 6 𝐸 = (mExβ€˜π‘‡)
5 elmsubrn.o . . . . . 6 𝑂 = (mRSubstβ€˜π‘‡)
61, 2, 3, 4, 5msubffval 35166 . . . . 5 (𝑇 ∈ V β†’ 𝑆 = (𝑔 ∈ ((mRExβ€˜π‘‡) ↑pm (mVRβ€˜π‘‡)) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘”)β€˜(2nd β€˜π‘’))⟩)))
71, 2, 5mrsubff 35155 . . . . . . . 8 (𝑇 ∈ V β†’ 𝑂:((mRExβ€˜π‘‡) ↑pm (mVRβ€˜π‘‡))⟢((mRExβ€˜π‘‡) ↑m (mRExβ€˜π‘‡)))
87ffnd 6728 . . . . . . 7 (𝑇 ∈ V β†’ 𝑂 Fn ((mRExβ€˜π‘‡) ↑pm (mVRβ€˜π‘‡)))
9 fnfvelrn 7095 . . . . . . 7 ((𝑂 Fn ((mRExβ€˜π‘‡) ↑pm (mVRβ€˜π‘‡)) ∧ 𝑔 ∈ ((mRExβ€˜π‘‡) ↑pm (mVRβ€˜π‘‡))) β†’ (π‘‚β€˜π‘”) ∈ ran 𝑂)
108, 9sylan 578 . . . . . 6 ((𝑇 ∈ V ∧ 𝑔 ∈ ((mRExβ€˜π‘‡) ↑pm (mVRβ€˜π‘‡))) β†’ (π‘‚β€˜π‘”) ∈ ran 𝑂)
117feqmptd 6972 . . . . . 6 (𝑇 ∈ V β†’ 𝑂 = (𝑔 ∈ ((mRExβ€˜π‘‡) ↑pm (mVRβ€˜π‘‡)) ↦ (π‘‚β€˜π‘”)))
12 eqidd 2729 . . . . . 6 (𝑇 ∈ V β†’ (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)) = (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)))
13 fveq1 6901 . . . . . . . 8 (𝑓 = (π‘‚β€˜π‘”) β†’ (π‘“β€˜(2nd β€˜π‘’)) = ((π‘‚β€˜π‘”)β€˜(2nd β€˜π‘’)))
1413opeq2d 4885 . . . . . . 7 (𝑓 = (π‘‚β€˜π‘”) β†’ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩ = ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘”)β€˜(2nd β€˜π‘’))⟩)
1514mpteq2dv 5254 . . . . . 6 (𝑓 = (π‘‚β€˜π‘”) β†’ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩) = (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘”)β€˜(2nd β€˜π‘’))⟩))
1610, 11, 12, 15fmptco 7144 . . . . 5 (𝑇 ∈ V β†’ ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)) ∘ 𝑂) = (𝑔 ∈ ((mRExβ€˜π‘‡) ↑pm (mVRβ€˜π‘‡)) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘”)β€˜(2nd β€˜π‘’))⟩)))
176, 16eqtr4d 2771 . . . 4 (𝑇 ∈ V β†’ 𝑆 = ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)) ∘ 𝑂))
1817rneqd 5944 . . 3 (𝑇 ∈ V β†’ ran 𝑆 = ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)) ∘ 𝑂))
19 rnco 6261 . . . 4 ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)) ∘ 𝑂) = ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)) β†Ύ ran 𝑂)
20 ssid 4004 . . . . . 6 ran 𝑂 βŠ† ran 𝑂
21 resmpt 6046 . . . . . 6 (ran 𝑂 βŠ† ran 𝑂 β†’ ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)) β†Ύ ran 𝑂) = (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)))
2220, 21ax-mp 5 . . . . 5 ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)) β†Ύ ran 𝑂) = (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩))
2322rneqi 5943 . . . 4 ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)) β†Ύ ran 𝑂) = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩))
2419, 23eqtri 2756 . . 3 ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)) ∘ 𝑂) = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩))
2518, 24eqtrdi 2784 . 2 (𝑇 ∈ V β†’ ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)))
26 mpt0 6702 . . . . 5 (𝑓 ∈ βˆ… ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)) = βˆ…
2726eqcomi 2737 . . . 4 βˆ… = (𝑓 ∈ βˆ… ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩))
28 fvprc 6894 . . . . 5 (Β¬ 𝑇 ∈ V β†’ (mSubstβ€˜π‘‡) = βˆ…)
293, 28eqtrid 2780 . . . 4 (Β¬ 𝑇 ∈ V β†’ 𝑆 = βˆ…)
305rnfvprc 6896 . . . . 5 (Β¬ 𝑇 ∈ V β†’ ran 𝑂 = βˆ…)
3130mpteq1d 5247 . . . 4 (Β¬ 𝑇 ∈ V β†’ (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)) = (𝑓 ∈ βˆ… ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)))
3227, 29, 313eqtr4a 2794 . . 3 (Β¬ 𝑇 ∈ V β†’ 𝑆 = (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)))
3332rneqd 5944 . 2 (Β¬ 𝑇 ∈ V β†’ ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)))
3425, 33pm2.61i 182 1 ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   = wceq 1533   ∈ wcel 2098  Vcvv 3473   βŠ† wss 3949  βˆ…c0 4326  βŸ¨cop 4638   ↦ cmpt 5235  ran crn 5683   β†Ύ cres 5684   ∘ ccom 5686   Fn wfn 6548  β€˜cfv 6553  (class class class)co 7426  1st c1st 7997  2nd c2nd 7998   ↑m cmap 8851   ↑pm cpm 8852  mVRcmvar 35104  mRExcmrex 35109  mExcmex 35110  mRSubstcmrsub 35113  mSubstcmsub 35114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-pm 8854  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-n0 12511  df-z 12597  df-uz 12861  df-fz 13525  df-fzo 13668  df-seq 14007  df-hash 14330  df-word 14505  df-concat 14561  df-s1 14586  df-struct 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-0g 17430  df-gsum 17431  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-submnd 18748  df-frmd 18808  df-mrex 35129  df-mrsub 35133  df-msub 35134
This theorem is referenced by:  msubco  35174  msubvrs  35203
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