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Theorem msubrn 32794
Description: Although it is defined for partial mappings of variables, every partial substitution is a substitution on some complete mapping of the variables. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
msubff.v 𝑉 = (mVR‘𝑇)
msubff.r 𝑅 = (mREx‘𝑇)
msubff.s 𝑆 = (mSubst‘𝑇)
Assertion
Ref Expression
msubrn ran 𝑆 = (𝑆 “ (𝑅m 𝑉))

Proof of Theorem msubrn
Dummy variables 𝑒 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 msubff.v . . . . . 6 𝑉 = (mVR‘𝑇)
2 msubff.r . . . . . 6 𝑅 = (mREx‘𝑇)
3 msubff.s . . . . . 6 𝑆 = (mSubst‘𝑇)
4 eqid 2824 . . . . . 6 (mEx‘𝑇) = (mEx‘𝑇)
5 eqid 2824 . . . . . 6 (mRSubst‘𝑇) = (mRSubst‘𝑇)
61, 2, 3, 4, 5msubffval 32788 . . . . 5 (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)))
76rneqd 5789 . . . 4 (𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)))
81, 2, 5mrsubff 32777 . . . . . . . . . 10 (𝑇 ∈ V → (mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅m 𝑅))
98adantr 484 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅m 𝑅))
109ffund 6499 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → Fun (mRSubst‘𝑇))
118ffnd 6496 . . . . . . . . . 10 (𝑇 ∈ V → (mRSubst‘𝑇) Fn (𝑅pm 𝑉))
12 fnfvelrn 6829 . . . . . . . . . 10 (((mRSubst‘𝑇) Fn (𝑅pm 𝑉) ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
1311, 12sylan 583 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
141, 2, 5mrsubrn 32778 . . . . . . . . 9 ran (mRSubst‘𝑇) = ((mRSubst‘𝑇) “ (𝑅m 𝑉))
1513, 14eleqtrdi 2926 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ((mRSubst‘𝑇) “ (𝑅m 𝑉)))
16 fvelima 6712 . . . . . . . 8 ((Fun (mRSubst‘𝑇) ∧ ((mRSubst‘𝑇)‘𝑓) ∈ ((mRSubst‘𝑇) “ (𝑅m 𝑉))) → ∃𝑔 ∈ (𝑅m 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓))
1710, 15, 16syl2anc 587 . . . . . . 7 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ∃𝑔 ∈ (𝑅m 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓))
18 elmapi 8411 . . . . . . . . . . . . 13 (𝑔 ∈ (𝑅m 𝑉) → 𝑔:𝑉𝑅)
1918adantl 485 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅m 𝑉)) → 𝑔:𝑉𝑅)
20 ssid 3973 . . . . . . . . . . . 12 𝑉𝑉
211, 2, 3, 4, 5msubfval 32789 . . . . . . . . . . . 12 ((𝑔:𝑉𝑅𝑉𝑉) → (𝑆𝑔) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩))
2219, 20, 21sylancl 589 . . . . . . . . . . 11 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅m 𝑉)) → (𝑆𝑔) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩))
23 fvex 6664 . . . . . . . . . . . . . . . 16 (mEx‘𝑇) ∈ V
2423mptex 6967 . . . . . . . . . . . . . . 15 (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩) ∈ V
25 eqid 2824 . . . . . . . . . . . . . . 15 (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)) = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩))
2624, 25fnmpti 6472 . . . . . . . . . . . . . 14 (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)) Fn (𝑅pm 𝑉)
276fneq1d 6427 . . . . . . . . . . . . . 14 (𝑇 ∈ V → (𝑆 Fn (𝑅pm 𝑉) ↔ (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)) Fn (𝑅pm 𝑉)))
2826, 27mpbiri 261 . . . . . . . . . . . . 13 (𝑇 ∈ V → 𝑆 Fn (𝑅pm 𝑉))
2928adantr 484 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅m 𝑉)) → 𝑆 Fn (𝑅pm 𝑉))
30 mapsspm 8423 . . . . . . . . . . . . 13 (𝑅m 𝑉) ⊆ (𝑅pm 𝑉)
3130a1i 11 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅m 𝑉)) → (𝑅m 𝑉) ⊆ (𝑅pm 𝑉))
32 simpr 488 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅m 𝑉)) → 𝑔 ∈ (𝑅m 𝑉))
33 fnfvima 6977 . . . . . . . . . . . 12 ((𝑆 Fn (𝑅pm 𝑉) ∧ (𝑅m 𝑉) ⊆ (𝑅pm 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉)) → (𝑆𝑔) ∈ (𝑆 “ (𝑅m 𝑉)))
3429, 31, 32, 33syl3anc 1368 . . . . . . . . . . 11 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅m 𝑉)) → (𝑆𝑔) ∈ (𝑆 “ (𝑅m 𝑉)))
3522, 34eqeltrrd 2917 . . . . . . . . . 10 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅m 𝑉)) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩) ∈ (𝑆 “ (𝑅m 𝑉)))
3635adantlr 714 . . . . . . . . 9 (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑔 ∈ (𝑅m 𝑉)) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩) ∈ (𝑆 “ (𝑅m 𝑉)))
37 fveq1 6650 . . . . . . . . . . . 12 (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒)) = (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒)))
3837opeq2d 4791 . . . . . . . . . . 11 (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩ = ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)
3938mpteq2dv 5143 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩))
4039eleq1d 2900 . . . . . . . . 9 (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → ((𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩) ∈ (𝑆 “ (𝑅m 𝑉)) ↔ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩) ∈ (𝑆 “ (𝑅m 𝑉))))
4136, 40syl5ibcom 248 . . . . . . . 8 (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑔 ∈ (𝑅m 𝑉)) → (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩) ∈ (𝑆 “ (𝑅m 𝑉))))
4241rexlimdva 3276 . . . . . . 7 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (∃𝑔 ∈ (𝑅m 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩) ∈ (𝑆 “ (𝑅m 𝑉))))
4317, 42mpd 15 . . . . . 6 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩) ∈ (𝑆 “ (𝑅m 𝑉)))
4443fmpttd 6860 . . . . 5 (𝑇 ∈ V → (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)):(𝑅pm 𝑉)⟶(𝑆 “ (𝑅m 𝑉)))
4544frnd 6502 . . . 4 (𝑇 ∈ V → ran (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)) ⊆ (𝑆 “ (𝑅m 𝑉)))
467, 45eqsstrd 3989 . . 3 (𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅m 𝑉)))
473rnfvprc 6645 . . . 4 𝑇 ∈ V → ran 𝑆 = ∅)
48 0ss 4331 . . . 4 ∅ ⊆ (𝑆 “ (𝑅m 𝑉))
4947, 48eqsstrdi 4005 . . 3 𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅m 𝑉)))
5046, 49pm2.61i 185 . 2 ran 𝑆 ⊆ (𝑆 “ (𝑅m 𝑉))
51 imassrn 5921 . 2 (𝑆 “ (𝑅m 𝑉)) ⊆ ran 𝑆
5250, 51eqssi 3967 1 ran 𝑆 = (𝑆 “ (𝑅m 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399   = wceq 1538  wcel 2115  wrex 3133  Vcvv 3479  wss 3918  c0 4274  cop 4554  cmpt 5127  ran crn 5537  cima 5539  Fun wfun 6330   Fn wfn 6331  wf 6332  cfv 6336  (class class class)co 7138  1st c1st 7670  2nd c2nd 7671  m cmap 8389  pm cpm 8390  mVRcmvar 32726  mRExcmrex 32731  mExcmex 32732  mRSubstcmrsub 32735  mSubstcmsub 32736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-cnex 10578  ax-resscn 10579  ax-1cn 10580  ax-icn 10581  ax-addcl 10582  ax-addrcl 10583  ax-mulcl 10584  ax-mulrcl 10585  ax-mulcom 10586  ax-addass 10587  ax-mulass 10588  ax-distr 10589  ax-i2m1 10590  ax-1ne0 10591  ax-1rid 10592  ax-rnegex 10593  ax-rrecex 10594  ax-cnre 10595  ax-pre-lttri 10596  ax-pre-lttrn 10597  ax-pre-ltadd 10598  ax-pre-mulgt0 10599
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-int 4858  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7096  df-ov 7141  df-oprab 7142  df-mpo 7143  df-om 7564  df-1st 7672  df-2nd 7673  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-card 9352  df-pnf 10662  df-mnf 10663  df-xr 10664  df-ltxr 10665  df-le 10666  df-sub 10857  df-neg 10858  df-nn 11624  df-2 11686  df-n0 11884  df-z 11968  df-uz 12230  df-fz 12884  df-fzo 13027  df-seq 13363  df-hash 13685  df-word 13856  df-concat 13912  df-s1 13939  df-struct 16474  df-ndx 16475  df-slot 16476  df-base 16478  df-sets 16479  df-ress 16480  df-plusg 16567  df-0g 16704  df-gsum 16705  df-mgm 17841  df-sgrp 17890  df-mnd 17901  df-submnd 17946  df-frmd 18003  df-mrex 32751  df-mrsub 32755  df-msub 32756
This theorem is referenced by:  msubff1o  32822
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