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Theorem msubrn 31805
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 2764 . . . . . 6 (mEx‘𝑇) = (mEx‘𝑇)
5 eqid 2764 . . . . . 6 (mRSubst‘𝑇) = (mRSubst‘𝑇)
61, 2, 3, 4, 5msubffval 31799 . . . . 5 (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)))
76rneqd 5520 . . . 4 (𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)))
81, 2, 5mrsubff 31788 . . . . . . . . . 10 (𝑇 ∈ V → (mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅))
98adantr 472 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅))
109ffund 6226 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → Fun (mRSubst‘𝑇))
118ffnd 6223 . . . . . . . . . 10 (𝑇 ∈ V → (mRSubst‘𝑇) Fn (𝑅pm 𝑉))
12 fnfvelrn 6545 . . . . . . . . . 10 (((mRSubst‘𝑇) Fn (𝑅pm 𝑉) ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
1311, 12sylan 575 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
141, 2, 5mrsubrn 31789 . . . . . . . . 9 ran (mRSubst‘𝑇) = ((mRSubst‘𝑇) “ (𝑅𝑚 𝑉))
1513, 14syl6eleq 2853 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ((mRSubst‘𝑇) “ (𝑅𝑚 𝑉)))
16 fvelima 6436 . . . . . . . 8 ((Fun (mRSubst‘𝑇) ∧ ((mRSubst‘𝑇)‘𝑓) ∈ ((mRSubst‘𝑇) “ (𝑅𝑚 𝑉))) → ∃𝑔 ∈ (𝑅𝑚 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓))
1710, 15, 16syl2anc 579 . . . . . . 7 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ∃𝑔 ∈ (𝑅𝑚 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓))
18 elmapi 8081 . . . . . . . . . . . . 13 (𝑔 ∈ (𝑅𝑚 𝑉) → 𝑔:𝑉𝑅)
1918adantl 473 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → 𝑔:𝑉𝑅)
20 ssid 3782 . . . . . . . . . . . 12 𝑉𝑉
211, 2, 3, 4, 5msubfval 31800 . . . . . . . . . . . 12 ((𝑔:𝑉𝑅𝑉𝑉) → (𝑆𝑔) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩))
2219, 20, 21sylancl 580 . . . . . . . . . . 11 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑆𝑔) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩))
23 fvex 6387 . . . . . . . . . . . . . . . 16 (mEx‘𝑇) ∈ V
2423mptex 6678 . . . . . . . . . . . . . . 15 (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩) ∈ V
25 eqid 2764 . . . . . . . . . . . . . . 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 6199 . . . . . . . . . . . . . 14 (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)) Fn (𝑅pm 𝑉)
276fneq1d 6158 . . . . . . . . . . . . . 14 (𝑇 ∈ V → (𝑆 Fn (𝑅pm 𝑉) ↔ (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)) Fn (𝑅pm 𝑉)))
2826, 27mpbiri 249 . . . . . . . . . . . . 13 (𝑇 ∈ V → 𝑆 Fn (𝑅pm 𝑉))
2928adantr 472 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → 𝑆 Fn (𝑅pm 𝑉))
30 mapsspm 8093 . . . . . . . . . . . . 13 (𝑅𝑚 𝑉) ⊆ (𝑅pm 𝑉)
3130a1i 11 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑅𝑚 𝑉) ⊆ (𝑅pm 𝑉))
32 simpr 477 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → 𝑔 ∈ (𝑅𝑚 𝑉))
33 fnfvima 6688 . . . . . . . . . . . 12 ((𝑆 Fn (𝑅pm 𝑉) ∧ (𝑅𝑚 𝑉) ⊆ (𝑅pm 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑆𝑔) ∈ (𝑆 “ (𝑅𝑚 𝑉)))
3429, 31, 32, 33syl3anc 1490 . . . . . . . . . . 11 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑆𝑔) ∈ (𝑆 “ (𝑅𝑚 𝑉)))
3522, 34eqeltrrd 2844 . . . . . . . . . 10 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩) ∈ (𝑆 “ (𝑅𝑚 𝑉)))
3635adantlr 706 . . . . . . . . 9 (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩) ∈ (𝑆 “ (𝑅𝑚 𝑉)))
37 fveq1 6373 . . . . . . . . . . . 12 (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒)) = (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒)))
3837opeq2d 4565 . . . . . . . . . . 11 (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩ = ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)
3938mpteq2dv 4903 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩))
4039eleq1d 2828 . . . . . . . . 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 236 . . . . . . . 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 3177 . . . . . . 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 6574 . . . . 5 (𝑇 ∈ V → (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)):(𝑅pm 𝑉)⟶(𝑆 “ (𝑅𝑚 𝑉)))
4544frnd 6229 . . . 4 (𝑇 ∈ V → ran (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)) ⊆ (𝑆 “ (𝑅𝑚 𝑉)))
467, 45eqsstrd 3798 . . 3 (𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅𝑚 𝑉)))
473rnfvprc 6368 . . . 4 𝑇 ∈ V → ran 𝑆 = ∅)
48 0ss 4133 . . . 4 ∅ ⊆ (𝑆 “ (𝑅𝑚 𝑉))
4947, 48syl6eqss 3814 . . 3 𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅𝑚 𝑉)))
5046, 49pm2.61i 176 . 2 ran 𝑆 ⊆ (𝑆 “ (𝑅𝑚 𝑉))
51 imassrn 5658 . 2 (𝑆 “ (𝑅𝑚 𝑉)) ⊆ ran 𝑆
5250, 51eqssi 3776 1 ran 𝑆 = (𝑆 “ (𝑅𝑚 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1652  wcel 2155  wrex 3055  Vcvv 3349  wss 3731  c0 4078  cop 4339  cmpt 4887  ran crn 5277  cima 5279  Fun wfun 6061   Fn wfn 6062  wf 6063  cfv 6067  (class class class)co 6841  1st c1st 7363  2nd c2nd 7364  𝑚 cmap 8059  pm cpm 8060  mVRcmvar 31737  mRExcmrex 31742  mExcmex 31743  mRSubstcmrsub 31746  mSubstcmsub 31747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146  ax-cnex 10244  ax-resscn 10245  ax-1cn 10246  ax-icn 10247  ax-addcl 10248  ax-addrcl 10249  ax-mulcl 10250  ax-mulrcl 10251  ax-mulcom 10252  ax-addass 10253  ax-mulass 10254  ax-distr 10255  ax-i2m1 10256  ax-1ne0 10257  ax-1rid 10258  ax-rnegex 10259  ax-rrecex 10260  ax-cnre 10261  ax-pre-lttri 10262  ax-pre-lttrn 10263  ax-pre-ltadd 10264  ax-pre-mulgt0 10265
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-int 4633  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-om 7263  df-1st 7365  df-2nd 7366  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-1o 7763  df-oadd 7767  df-er 7946  df-map 8061  df-pm 8062  df-en 8160  df-dom 8161  df-sdom 8162  df-fin 8163  df-card 9015  df-pnf 10329  df-mnf 10330  df-xr 10331  df-ltxr 10332  df-le 10333  df-sub 10521  df-neg 10522  df-nn 11274  df-2 11334  df-n0 11538  df-z 11624  df-uz 11886  df-fz 12533  df-fzo 12673  df-seq 13008  df-hash 13321  df-word 13486  df-concat 13541  df-s1 13566  df-struct 16133  df-ndx 16134  df-slot 16135  df-base 16137  df-sets 16138  df-ress 16139  df-plusg 16228  df-0g 16369  df-gsum 16370  df-mgm 17509  df-sgrp 17551  df-mnd 17562  df-submnd 17603  df-frmd 17654  df-mrex 31762  df-mrsub 31766  df-msub 31767
This theorem is referenced by:  msubff1o  31833
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