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Theorem elmsubrn 34186
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 2733 . . . . . 6 (mVRβ€˜π‘‡) = (mVRβ€˜π‘‡)
2 eqid 2733 . . . . . 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 34181 . . . . 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 34170 . . . . . . . 8 (𝑇 ∈ V β†’ 𝑂:((mRExβ€˜π‘‡) ↑pm (mVRβ€˜π‘‡))⟢((mRExβ€˜π‘‡) ↑m (mRExβ€˜π‘‡)))
87ffnd 6673 . . . . . . 7 (𝑇 ∈ V β†’ 𝑂 Fn ((mRExβ€˜π‘‡) ↑pm (mVRβ€˜π‘‡)))
9 fnfvelrn 7035 . . . . . . 7 ((𝑂 Fn ((mRExβ€˜π‘‡) ↑pm (mVRβ€˜π‘‡)) ∧ 𝑔 ∈ ((mRExβ€˜π‘‡) ↑pm (mVRβ€˜π‘‡))) β†’ (π‘‚β€˜π‘”) ∈ ran 𝑂)
108, 9sylan 581 . . . . . 6 ((𝑇 ∈ V ∧ 𝑔 ∈ ((mRExβ€˜π‘‡) ↑pm (mVRβ€˜π‘‡))) β†’ (π‘‚β€˜π‘”) ∈ ran 𝑂)
117feqmptd 6914 . . . . . 6 (𝑇 ∈ V β†’ 𝑂 = (𝑔 ∈ ((mRExβ€˜π‘‡) ↑pm (mVRβ€˜π‘‡)) ↦ (π‘‚β€˜π‘”)))
12 eqidd 2734 . . . . . 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 6845 . . . . . . . 8 (𝑓 = (π‘‚β€˜π‘”) β†’ (π‘“β€˜(2nd β€˜π‘’)) = ((π‘‚β€˜π‘”)β€˜(2nd β€˜π‘’)))
1413opeq2d 4841 . . . . . . 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 5211 . . . . . 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 7079 . . . . 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 2776 . . . 4 (𝑇 ∈ V β†’ 𝑆 = ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)) ∘ 𝑂))
1817rneqd 5897 . . 3 (𝑇 ∈ V β†’ ran 𝑆 = ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)) ∘ 𝑂))
19 rnco 6208 . . . 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 3970 . . . . . 6 ran 𝑂 βŠ† ran 𝑂
21 resmpt 5995 . . . . . 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 5896 . . . 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 2761 . . 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 2789 . 2 (𝑇 ∈ V β†’ ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)))
26 mpt0 6647 . . . . 5 (𝑓 ∈ βˆ… ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)) = βˆ…
2726eqcomi 2742 . . . 4 βˆ… = (𝑓 ∈ βˆ… ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩))
28 fvprc 6838 . . . . 5 (Β¬ 𝑇 ∈ V β†’ (mSubstβ€˜π‘‡) = βˆ…)
293, 28eqtrid 2785 . . . 4 (Β¬ 𝑇 ∈ V β†’ 𝑆 = βˆ…)
305rnfvprc 6840 . . . . 5 (Β¬ 𝑇 ∈ V β†’ ran 𝑂 = βˆ…)
3130mpteq1d 5204 . . . 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 2799 . . 3 (Β¬ 𝑇 ∈ V β†’ 𝑆 = (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), (π‘“β€˜(2nd β€˜π‘’))⟩)))
3332rneqd 5897 . 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 1542   ∈ wcel 2107  Vcvv 3447   βŠ† wss 3914  βˆ…c0 4286  βŸ¨cop 4596   ↦ cmpt 5192  ran crn 5638   β†Ύ cres 5639   ∘ ccom 5641   Fn wfn 6495  β€˜cfv 6500  (class class class)co 7361  1st c1st 7923  2nd c2nd 7924   ↑m cmap 8771   ↑pm cpm 8772  mVRcmvar 34119  mRExcmrex 34124  mExcmex 34125  mRSubstcmrsub 34128  mSubstcmsub 34129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-pm 8774  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-n0 12422  df-z 12508  df-uz 12772  df-fz 13434  df-fzo 13577  df-seq 13916  df-hash 14240  df-word 14412  df-concat 14468  df-s1 14493  df-struct 17027  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-0g 17331  df-gsum 17332  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-submnd 18610  df-frmd 18667  df-mrex 34144  df-mrsub 34148  df-msub 34149
This theorem is referenced by:  msubco  34189  msubvrs  34218
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