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Theorem msubrn 34551
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 = (mVRβ€˜π‘‡)
msubff.r 𝑅 = (mRExβ€˜π‘‡)
msubff.s 𝑆 = (mSubstβ€˜π‘‡)
Assertion
Ref Expression
msubrn ran 𝑆 = (𝑆 β€œ (𝑅 ↑m 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 = (mVRβ€˜π‘‡)
2 msubff.r . . . . . 6 𝑅 = (mRExβ€˜π‘‡)
3 msubff.s . . . . . 6 𝑆 = (mSubstβ€˜π‘‡)
4 eqid 2733 . . . . . 6 (mExβ€˜π‘‡) = (mExβ€˜π‘‡)
5 eqid 2733 . . . . . 6 (mRSubstβ€˜π‘‡) = (mRSubstβ€˜π‘‡)
61, 2, 3, 4, 5msubffval 34545 . . . . 5 (𝑇 ∈ V β†’ 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)))
76rneqd 5938 . . . 4 (𝑇 ∈ V β†’ ran 𝑆 = ran (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)))
81, 2, 5mrsubff 34534 . . . . . . . . . 10 (𝑇 ∈ V β†’ (mRSubstβ€˜π‘‡):(𝑅 ↑pm 𝑉)⟢(𝑅 ↑m 𝑅))
98adantr 482 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) β†’ (mRSubstβ€˜π‘‡):(𝑅 ↑pm 𝑉)⟢(𝑅 ↑m 𝑅))
109ffund 6722 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) β†’ Fun (mRSubstβ€˜π‘‡))
118ffnd 6719 . . . . . . . . . 10 (𝑇 ∈ V β†’ (mRSubstβ€˜π‘‡) Fn (𝑅 ↑pm 𝑉))
12 fnfvelrn 7083 . . . . . . . . . 10 (((mRSubstβ€˜π‘‡) Fn (𝑅 ↑pm 𝑉) ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) β†’ ((mRSubstβ€˜π‘‡)β€˜π‘“) ∈ ran (mRSubstβ€˜π‘‡))
1311, 12sylan 581 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) β†’ ((mRSubstβ€˜π‘‡)β€˜π‘“) ∈ ran (mRSubstβ€˜π‘‡))
141, 2, 5mrsubrn 34535 . . . . . . . . 9 ran (mRSubstβ€˜π‘‡) = ((mRSubstβ€˜π‘‡) β€œ (𝑅 ↑m 𝑉))
1513, 14eleqtrdi 2844 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) β†’ ((mRSubstβ€˜π‘‡)β€˜π‘“) ∈ ((mRSubstβ€˜π‘‡) β€œ (𝑅 ↑m 𝑉)))
16 fvelima 6958 . . . . . . . 8 ((Fun (mRSubstβ€˜π‘‡) ∧ ((mRSubstβ€˜π‘‡)β€˜π‘“) ∈ ((mRSubstβ€˜π‘‡) β€œ (𝑅 ↑m 𝑉))) β†’ βˆƒπ‘” ∈ (𝑅 ↑m 𝑉)((mRSubstβ€˜π‘‡)β€˜π‘”) = ((mRSubstβ€˜π‘‡)β€˜π‘“))
1710, 15, 16syl2anc 585 . . . . . . 7 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) β†’ βˆƒπ‘” ∈ (𝑅 ↑m 𝑉)((mRSubstβ€˜π‘‡)β€˜π‘”) = ((mRSubstβ€˜π‘‡)β€˜π‘“))
18 elmapi 8843 . . . . . . . . . . . . 13 (𝑔 ∈ (𝑅 ↑m 𝑉) β†’ 𝑔:π‘‰βŸΆπ‘…)
1918adantl 483 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑m 𝑉)) β†’ 𝑔:π‘‰βŸΆπ‘…)
20 ssid 4005 . . . . . . . . . . . 12 𝑉 βŠ† 𝑉
211, 2, 3, 4, 5msubfval 34546 . . . . . . . . . . . 12 ((𝑔:π‘‰βŸΆπ‘… ∧ 𝑉 βŠ† 𝑉) β†’ (π‘†β€˜π‘”) = (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘”)β€˜(2nd β€˜π‘’))⟩))
2219, 20, 21sylancl 587 . . . . . . . . . . 11 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑m 𝑉)) β†’ (π‘†β€˜π‘”) = (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘”)β€˜(2nd β€˜π‘’))⟩))
23 fvex 6905 . . . . . . . . . . . . . . . 16 (mExβ€˜π‘‡) ∈ V
2423mptex 7225 . . . . . . . . . . . . . . 15 (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩) ∈ V
25 eqid 2733 . . . . . . . . . . . . . . 15 (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)) = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩))
2624, 25fnmpti 6694 . . . . . . . . . . . . . 14 (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)) Fn (𝑅 ↑pm 𝑉)
276fneq1d 6643 . . . . . . . . . . . . . 14 (𝑇 ∈ V β†’ (𝑆 Fn (𝑅 ↑pm 𝑉) ↔ (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)) Fn (𝑅 ↑pm 𝑉)))
2826, 27mpbiri 258 . . . . . . . . . . . . 13 (𝑇 ∈ V β†’ 𝑆 Fn (𝑅 ↑pm 𝑉))
2928adantr 482 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑m 𝑉)) β†’ 𝑆 Fn (𝑅 ↑pm 𝑉))
30 mapsspm 8870 . . . . . . . . . . . . 13 (𝑅 ↑m 𝑉) βŠ† (𝑅 ↑pm 𝑉)
3130a1i 11 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑m 𝑉)) β†’ (𝑅 ↑m 𝑉) βŠ† (𝑅 ↑pm 𝑉))
32 simpr 486 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑m 𝑉)) β†’ 𝑔 ∈ (𝑅 ↑m 𝑉))
33 fnfvima 7235 . . . . . . . . . . . 12 ((𝑆 Fn (𝑅 ↑pm 𝑉) ∧ (𝑅 ↑m 𝑉) βŠ† (𝑅 ↑pm 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉)) β†’ (π‘†β€˜π‘”) ∈ (𝑆 β€œ (𝑅 ↑m 𝑉)))
3429, 31, 32, 33syl3anc 1372 . . . . . . . . . . 11 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑m 𝑉)) β†’ (π‘†β€˜π‘”) ∈ (𝑆 β€œ (𝑅 ↑m 𝑉)))
3522, 34eqeltrrd 2835 . . . . . . . . . 10 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑m 𝑉)) β†’ (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘”)β€˜(2nd β€˜π‘’))⟩) ∈ (𝑆 β€œ (𝑅 ↑m 𝑉)))
3635adantlr 714 . . . . . . . . 9 (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉)) β†’ (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘”)β€˜(2nd β€˜π‘’))⟩) ∈ (𝑆 β€œ (𝑅 ↑m 𝑉)))
37 fveq1 6891 . . . . . . . . . . . 12 (((mRSubstβ€˜π‘‡)β€˜π‘”) = ((mRSubstβ€˜π‘‡)β€˜π‘“) β†’ (((mRSubstβ€˜π‘‡)β€˜π‘”)β€˜(2nd β€˜π‘’)) = (((mRSubstβ€˜π‘‡)β€˜π‘“)β€˜(2nd β€˜π‘’)))
3837opeq2d 4881 . . . . . . . . . . 11 (((mRSubstβ€˜π‘‡)β€˜π‘”) = ((mRSubstβ€˜π‘‡)β€˜π‘“) β†’ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘”)β€˜(2nd β€˜π‘’))⟩ = ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)
3938mpteq2dv 5251 . . . . . . . . . 10 (((mRSubstβ€˜π‘‡)β€˜π‘”) = ((mRSubstβ€˜π‘‡)β€˜π‘“) β†’ (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘”)β€˜(2nd β€˜π‘’))⟩) = (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩))
4039eleq1d 2819 . . . . . . . . 9 (((mRSubstβ€˜π‘‡)β€˜π‘”) = ((mRSubstβ€˜π‘‡)β€˜π‘“) β†’ ((𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘”)β€˜(2nd β€˜π‘’))⟩) ∈ (𝑆 β€œ (𝑅 ↑m 𝑉)) ↔ (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩) ∈ (𝑆 β€œ (𝑅 ↑m 𝑉))))
4136, 40syl5ibcom 244 . . . . . . . 8 (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉)) β†’ (((mRSubstβ€˜π‘‡)β€˜π‘”) = ((mRSubstβ€˜π‘‡)β€˜π‘“) β†’ (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩) ∈ (𝑆 β€œ (𝑅 ↑m 𝑉))))
4241rexlimdva 3156 . . . . . . 7 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) β†’ (βˆƒπ‘” ∈ (𝑅 ↑m 𝑉)((mRSubstβ€˜π‘‡)β€˜π‘”) = ((mRSubstβ€˜π‘‡)β€˜π‘“) β†’ (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩) ∈ (𝑆 β€œ (𝑅 ↑m 𝑉))))
4317, 42mpd 15 . . . . . 6 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) β†’ (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩) ∈ (𝑆 β€œ (𝑅 ↑m 𝑉)))
4443fmpttd 7115 . . . . 5 (𝑇 ∈ V β†’ (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)):(𝑅 ↑pm 𝑉)⟢(𝑆 β€œ (𝑅 ↑m 𝑉)))
4544frnd 6726 . . . 4 (𝑇 ∈ V β†’ ran (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)) βŠ† (𝑆 β€œ (𝑅 ↑m 𝑉)))
467, 45eqsstrd 4021 . . 3 (𝑇 ∈ V β†’ ran 𝑆 βŠ† (𝑆 β€œ (𝑅 ↑m 𝑉)))
473rnfvprc 6886 . . . 4 (Β¬ 𝑇 ∈ V β†’ ran 𝑆 = βˆ…)
48 0ss 4397 . . . 4 βˆ… βŠ† (𝑆 β€œ (𝑅 ↑m 𝑉))
4947, 48eqsstrdi 4037 . . 3 (Β¬ 𝑇 ∈ V β†’ ran 𝑆 βŠ† (𝑆 β€œ (𝑅 ↑m 𝑉)))
5046, 49pm2.61i 182 . 2 ran 𝑆 βŠ† (𝑆 β€œ (𝑅 ↑m 𝑉))
51 imassrn 6071 . 2 (𝑆 β€œ (𝑅 ↑m 𝑉)) βŠ† ran 𝑆
5250, 51eqssi 3999 1 ran 𝑆 = (𝑆 β€œ (𝑅 ↑m 𝑉))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071  Vcvv 3475   βŠ† wss 3949  βˆ…c0 4323  βŸ¨cop 4635   ↦ cmpt 5232  ran crn 5678   β€œ cima 5680  Fun wfun 6538   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974   ↑m cmap 8820   ↑pm cpm 8821  mVRcmvar 34483  mRExcmrex 34488  mExcmex 34489  mRSubstcmrsub 34492  mSubstcmsub 34493
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-seq 13967  df-hash 14291  df-word 14465  df-concat 14521  df-s1 14546  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-0g 17387  df-gsum 17388  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-submnd 18672  df-frmd 18730  df-mrex 34508  df-mrsub 34512  df-msub 34513
This theorem is referenced by:  msubff1o  34579
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