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Theorem elmsubrn 35046
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 2726 . . . . . 6 (mVRβ€˜π‘‡) = (mVRβ€˜π‘‡)
2 eqid 2726 . . . . . 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 35041 . . . . 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 35030 . . . . . . . 8 (𝑇 ∈ V β†’ 𝑂:((mRExβ€˜π‘‡) ↑pm (mVRβ€˜π‘‡))⟢((mRExβ€˜π‘‡) ↑m (mRExβ€˜π‘‡)))
87ffnd 6711 . . . . . . 7 (𝑇 ∈ V β†’ 𝑂 Fn ((mRExβ€˜π‘‡) ↑pm (mVRβ€˜π‘‡)))
9 fnfvelrn 7075 . . . . . . 7 ((𝑂 Fn ((mRExβ€˜π‘‡) ↑pm (mVRβ€˜π‘‡)) ∧ 𝑔 ∈ ((mRExβ€˜π‘‡) ↑pm (mVRβ€˜π‘‡))) β†’ (π‘‚β€˜π‘”) ∈ ran 𝑂)
108, 9sylan 579 . . . . . 6 ((𝑇 ∈ V ∧ 𝑔 ∈ ((mRExβ€˜π‘‡) ↑pm (mVRβ€˜π‘‡))) β†’ (π‘‚β€˜π‘”) ∈ ran 𝑂)
117feqmptd 6953 . . . . . 6 (𝑇 ∈ V β†’ 𝑂 = (𝑔 ∈ ((mRExβ€˜π‘‡) ↑pm (mVRβ€˜π‘‡)) ↦ (π‘‚β€˜π‘”)))
12 eqidd 2727 . . . . . 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 6883 . . . . . . . 8 (𝑓 = (π‘‚β€˜π‘”) β†’ (π‘“β€˜(2nd β€˜π‘’)) = ((π‘‚β€˜π‘”)β€˜(2nd β€˜π‘’)))
1413opeq2d 4875 . . . . . . 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 5243 . . . . . 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 7122 . . . . 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 2769 . . . 4 (𝑇 ∈ V β†’ 𝑆 = ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)) ∘ 𝑂))
1817rneqd 5930 . . 3 (𝑇 ∈ V β†’ ran 𝑆 = ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)) ∘ 𝑂))
19 rnco 6244 . . . 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 3999 . . . . . 6 ran 𝑂 βŠ† ran 𝑂
21 resmpt 6030 . . . . . 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 5929 . . . 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 2754 . . 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 2782 . 2 (𝑇 ∈ V β†’ ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)))
26 mpt0 6685 . . . . 5 (𝑓 ∈ βˆ… ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)) = βˆ…
2726eqcomi 2735 . . . 4 βˆ… = (𝑓 ∈ βˆ… ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩))
28 fvprc 6876 . . . . 5 (Β¬ 𝑇 ∈ V β†’ (mSubstβ€˜π‘‡) = βˆ…)
293, 28eqtrid 2778 . . . 4 (Β¬ 𝑇 ∈ V β†’ 𝑆 = βˆ…)
305rnfvprc 6878 . . . . 5 (Β¬ 𝑇 ∈ V β†’ ran 𝑂 = βˆ…)
3130mpteq1d 5236 . . . 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 2792 . . 3 (Β¬ 𝑇 ∈ V β†’ 𝑆 = (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)))
3332rneqd 5930 . 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 3468   βŠ† wss 3943  βˆ…c0 4317  βŸ¨cop 4629   ↦ cmpt 5224  ran crn 5670   β†Ύ cres 5671   ∘ ccom 5673   Fn wfn 6531  β€˜cfv 6536  (class class class)co 7404  1st c1st 7969  2nd c2nd 7970   ↑m cmap 8819   ↑pm cpm 8820  mVRcmvar 34979  mRExcmrex 34984  mExcmex 34985  mRSubstcmrsub 34988  mSubstcmsub 34989
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-er 8702  df-map 8821  df-pm 8822  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-card 9933  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-n0 12474  df-z 12560  df-uz 12824  df-fz 13488  df-fzo 13631  df-seq 13970  df-hash 14293  df-word 14468  df-concat 14524  df-s1 14549  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17151  df-ress 17180  df-plusg 17216  df-0g 17393  df-gsum 17394  df-mgm 18570  df-sgrp 18649  df-mnd 18665  df-submnd 18711  df-frmd 18771  df-mrex 35004  df-mrsub 35008  df-msub 35009
This theorem is referenced by:  msubco  35049  msubvrs  35078
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