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Theorem msubrn 34508
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 2732 . . . . . 6 (mExβ€˜π‘‡) = (mExβ€˜π‘‡)
5 eqid 2732 . . . . . 6 (mRSubstβ€˜π‘‡) = (mRSubstβ€˜π‘‡)
61, 2, 3, 4, 5msubffval 34502 . . . . 5 (𝑇 ∈ V β†’ 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)))
76rneqd 5935 . . . 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 34491 . . . . . . . . . 10 (𝑇 ∈ V β†’ (mRSubstβ€˜π‘‡):(𝑅 ↑pm 𝑉)⟢(𝑅 ↑m 𝑅))
98adantr 481 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) β†’ (mRSubstβ€˜π‘‡):(𝑅 ↑pm 𝑉)⟢(𝑅 ↑m 𝑅))
109ffund 6718 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) β†’ Fun (mRSubstβ€˜π‘‡))
118ffnd 6715 . . . . . . . . . 10 (𝑇 ∈ V β†’ (mRSubstβ€˜π‘‡) Fn (𝑅 ↑pm 𝑉))
12 fnfvelrn 7079 . . . . . . . . . 10 (((mRSubstβ€˜π‘‡) Fn (𝑅 ↑pm 𝑉) ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) β†’ ((mRSubstβ€˜π‘‡)β€˜π‘“) ∈ ran (mRSubstβ€˜π‘‡))
1311, 12sylan 580 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) β†’ ((mRSubstβ€˜π‘‡)β€˜π‘“) ∈ ran (mRSubstβ€˜π‘‡))
141, 2, 5mrsubrn 34492 . . . . . . . . 9 ran (mRSubstβ€˜π‘‡) = ((mRSubstβ€˜π‘‡) β€œ (𝑅 ↑m 𝑉))
1513, 14eleqtrdi 2843 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) β†’ ((mRSubstβ€˜π‘‡)β€˜π‘“) ∈ ((mRSubstβ€˜π‘‡) β€œ (𝑅 ↑m 𝑉)))
16 fvelima 6954 . . . . . . . 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 584 . . . . . . 7 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) β†’ βˆƒπ‘” ∈ (𝑅 ↑m 𝑉)((mRSubstβ€˜π‘‡)β€˜π‘”) = ((mRSubstβ€˜π‘‡)β€˜π‘“))
18 elmapi 8839 . . . . . . . . . . . . 13 (𝑔 ∈ (𝑅 ↑m 𝑉) β†’ 𝑔:π‘‰βŸΆπ‘…)
1918adantl 482 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑m 𝑉)) β†’ 𝑔:π‘‰βŸΆπ‘…)
20 ssid 4003 . . . . . . . . . . . 12 𝑉 βŠ† 𝑉
211, 2, 3, 4, 5msubfval 34503 . . . . . . . . . . . 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 586 . . . . . . . . . . 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 6901 . . . . . . . . . . . . . . . 16 (mExβ€˜π‘‡) ∈ V
2423mptex 7221 . . . . . . . . . . . . . . 15 (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩) ∈ V
25 eqid 2732 . . . . . . . . . . . . . . 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 6690 . . . . . . . . . . . . . 14 (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘’), (((mRSubstβ€˜π‘‡)β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)) Fn (𝑅 ↑pm 𝑉)
276fneq1d 6639 . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . 13 (𝑇 ∈ V β†’ 𝑆 Fn (𝑅 ↑pm 𝑉))
2928adantr 481 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑m 𝑉)) β†’ 𝑆 Fn (𝑅 ↑pm 𝑉))
30 mapsspm 8866 . . . . . . . . . . . . 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 485 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑m 𝑉)) β†’ 𝑔 ∈ (𝑅 ↑m 𝑉))
33 fnfvima 7231 . . . . . . . . . . . 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 1371 . . . . . . . . . . 11 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑m 𝑉)) β†’ (π‘†β€˜π‘”) ∈ (𝑆 β€œ (𝑅 ↑m 𝑉)))
3522, 34eqeltrrd 2834 . . . . . . . . . 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 713 . . . . . . . . 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 6887 . . . . . . . . . . . 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 4879 . . . . . . . . . . 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 5249 . . . . . . . . . 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 2818 . . . . . . . . 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 3155 . . . . . . 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 7111 . . . . 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 6722 . . . 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 4019 . . 3 (𝑇 ∈ V β†’ ran 𝑆 βŠ† (𝑆 β€œ (𝑅 ↑m 𝑉)))
473rnfvprc 6882 . . . 4 (Β¬ 𝑇 ∈ V β†’ ran 𝑆 = βˆ…)
48 0ss 4395 . . . 4 βˆ… βŠ† (𝑆 β€œ (𝑅 ↑m 𝑉))
4947, 48eqsstrdi 4035 . . 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 6068 . 2 (𝑆 β€œ (𝑅 ↑m 𝑉)) βŠ† ran 𝑆
5250, 51eqssi 3997 1 ran 𝑆 = (𝑆 β€œ (𝑅 ↑m 𝑉))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  Vcvv 3474   βŠ† wss 3947  βˆ…c0 4321  βŸ¨cop 4633   ↦ cmpt 5230  ran crn 5676   β€œ cima 5678  Fun wfun 6534   Fn wfn 6535  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405  1st c1st 7969  2nd c2nd 7970   ↑m cmap 8816   ↑pm cpm 8817  mVRcmvar 34440  mRExcmrex 34445  mExcmex 34446  mRSubstcmrsub 34449  mSubstcmsub 34450
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-seq 13963  df-hash 14287  df-word 14461  df-concat 14517  df-s1 14542  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-0g 17383  df-gsum 17384  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-submnd 18668  df-frmd 18726  df-mrex 34465  df-mrsub 34469  df-msub 34470
This theorem is referenced by:  msubff1o  34536
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