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Theorem msubrn 31972
 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 2825 . . . . . 6 (mEx‘𝑇) = (mEx‘𝑇)
5 eqid 2825 . . . . . 6 (mRSubst‘𝑇) = (mRSubst‘𝑇)
61, 2, 3, 4, 5msubffval 31966 . . . . 5 (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)))
76rneqd 5585 . . . 4 (𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)))
81, 2, 5mrsubff 31955 . . . . . . . . . 10 (𝑇 ∈ V → (mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅))
98adantr 474 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅))
109ffund 6282 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → Fun (mRSubst‘𝑇))
118ffnd 6279 . . . . . . . . . 10 (𝑇 ∈ V → (mRSubst‘𝑇) Fn (𝑅pm 𝑉))
12 fnfvelrn 6605 . . . . . . . . . 10 (((mRSubst‘𝑇) Fn (𝑅pm 𝑉) ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
1311, 12sylan 577 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
141, 2, 5mrsubrn 31956 . . . . . . . . 9 ran (mRSubst‘𝑇) = ((mRSubst‘𝑇) “ (𝑅𝑚 𝑉))
1513, 14syl6eleq 2916 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ((mRSubst‘𝑇) “ (𝑅𝑚 𝑉)))
16 fvelima 6495 . . . . . . . 8 ((Fun (mRSubst‘𝑇) ∧ ((mRSubst‘𝑇)‘𝑓) ∈ ((mRSubst‘𝑇) “ (𝑅𝑚 𝑉))) → ∃𝑔 ∈ (𝑅𝑚 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓))
1710, 15, 16syl2anc 581 . . . . . . 7 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ∃𝑔 ∈ (𝑅𝑚 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓))
18 elmapi 8144 . . . . . . . . . . . . 13 (𝑔 ∈ (𝑅𝑚 𝑉) → 𝑔:𝑉𝑅)
1918adantl 475 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → 𝑔:𝑉𝑅)
20 ssid 3848 . . . . . . . . . . . 12 𝑉𝑉
211, 2, 3, 4, 5msubfval 31967 . . . . . . . . . . . 12 ((𝑔:𝑉𝑅𝑉𝑉) → (𝑆𝑔) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩))
2219, 20, 21sylancl 582 . . . . . . . . . . 11 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑆𝑔) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩))
23 fvex 6446 . . . . . . . . . . . . . . . 16 (mEx‘𝑇) ∈ V
2423mptex 6742 . . . . . . . . . . . . . . 15 (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩) ∈ V
25 eqid 2825 . . . . . . . . . . . . . . 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 6255 . . . . . . . . . . . . . 14 (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)) Fn (𝑅pm 𝑉)
276fneq1d 6214 . . . . . . . . . . . . . 14 (𝑇 ∈ V → (𝑆 Fn (𝑅pm 𝑉) ↔ (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)) Fn (𝑅pm 𝑉)))
2826, 27mpbiri 250 . . . . . . . . . . . . 13 (𝑇 ∈ V → 𝑆 Fn (𝑅pm 𝑉))
2928adantr 474 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → 𝑆 Fn (𝑅pm 𝑉))
30 mapsspm 8156 . . . . . . . . . . . . 13 (𝑅𝑚 𝑉) ⊆ (𝑅pm 𝑉)
3130a1i 11 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑅𝑚 𝑉) ⊆ (𝑅pm 𝑉))
32 simpr 479 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → 𝑔 ∈ (𝑅𝑚 𝑉))
33 fnfvima 6752 . . . . . . . . . . . 12 ((𝑆 Fn (𝑅pm 𝑉) ∧ (𝑅𝑚 𝑉) ⊆ (𝑅pm 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑆𝑔) ∈ (𝑆 “ (𝑅𝑚 𝑉)))
3429, 31, 32, 33syl3anc 1496 . . . . . . . . . . 11 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑆𝑔) ∈ (𝑆 “ (𝑅𝑚 𝑉)))
3522, 34eqeltrrd 2907 . . . . . . . . . 10 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩) ∈ (𝑆 “ (𝑅𝑚 𝑉)))
3635adantlr 708 . . . . . . . . 9 (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩) ∈ (𝑆 “ (𝑅𝑚 𝑉)))
37 fveq1 6432 . . . . . . . . . . . 12 (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒)) = (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒)))
3837opeq2d 4630 . . . . . . . . . . 11 (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩ = ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)
3938mpteq2dv 4968 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩))
4039eleq1d 2891 . . . . . . . . 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 237 . . . . . . . 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 3240 . . . . . . 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 6634 . . . . 5 (𝑇 ∈ V → (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)):(𝑅pm 𝑉)⟶(𝑆 “ (𝑅𝑚 𝑉)))
4544frnd 6285 . . . 4 (𝑇 ∈ V → ran (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)) ⊆ (𝑆 “ (𝑅𝑚 𝑉)))
467, 45eqsstrd 3864 . . 3 (𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅𝑚 𝑉)))
473rnfvprc 6427 . . . 4 𝑇 ∈ V → ran 𝑆 = ∅)
48 0ss 4197 . . . 4 ∅ ⊆ (𝑆 “ (𝑅𝑚 𝑉))
4947, 48syl6eqss 3880 . . 3 𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅𝑚 𝑉)))
5046, 49pm2.61i 177 . 2 ran 𝑆 ⊆ (𝑆 “ (𝑅𝑚 𝑉))
51 imassrn 5718 . 2 (𝑆 “ (𝑅𝑚 𝑉)) ⊆ ran 𝑆
5250, 51eqssi 3843 1 ran 𝑆 = (𝑆 “ (𝑅𝑚 𝑉))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∃wrex 3118  Vcvv 3414   ⊆ wss 3798  ∅c0 4144  ⟨cop 4403   ↦ cmpt 4952  ran crn 5343   “ cima 5345  Fun wfun 6117   Fn wfn 6118  ⟶wf 6119  ‘cfv 6123  (class class class)co 6905  1st c1st 7426  2nd c2nd 7427   ↑𝑚 cmap 8122   ↑pm cpm 8123  mVRcmvar 31904  mRExcmrex 31909  mExcmex 31910  mRSubstcmrsub 31913  mSubstcmsub 31914 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-er 8009  df-map 8124  df-pm 8125  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-card 9078  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-n0 11619  df-z 11705  df-uz 11969  df-fz 12620  df-fzo 12761  df-seq 13096  df-hash 13411  df-word 13575  df-concat 13631  df-s1 13656  df-struct 16224  df-ndx 16225  df-slot 16226  df-base 16228  df-sets 16229  df-ress 16230  df-plusg 16318  df-0g 16455  df-gsum 16456  df-mgm 17595  df-sgrp 17637  df-mnd 17648  df-submnd 17689  df-frmd 17740  df-mrex 31929  df-mrsub 31933  df-msub 31934 This theorem is referenced by:  msubff1o  32000
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