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Theorem msubrn 31155
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 2621 . . . . . 6 (mEx‘𝑇) = (mEx‘𝑇)
5 eqid 2621 . . . . . 6 (mRSubst‘𝑇) = (mRSubst‘𝑇)
61, 2, 3, 4, 5msubffval 31149 . . . . 5 (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)))
76rneqd 5315 . . . 4 (𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)))
81, 2, 5mrsubff 31138 . . . . . . . . . 10 (𝑇 ∈ V → (mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅))
98adantr 481 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅))
10 ffun 6007 . . . . . . . . 9 ((mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅) → Fun (mRSubst‘𝑇))
119, 10syl 17 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → Fun (mRSubst‘𝑇))
12 ffn 6004 . . . . . . . . . . 11 ((mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅) → (mRSubst‘𝑇) Fn (𝑅pm 𝑉))
138, 12syl 17 . . . . . . . . . 10 (𝑇 ∈ V → (mRSubst‘𝑇) Fn (𝑅pm 𝑉))
14 fnfvelrn 6314 . . . . . . . . . 10 (((mRSubst‘𝑇) Fn (𝑅pm 𝑉) ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
1513, 14sylan 488 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
161, 2, 5mrsubrn 31139 . . . . . . . . 9 ran (mRSubst‘𝑇) = ((mRSubst‘𝑇) “ (𝑅𝑚 𝑉))
1715, 16syl6eleq 2708 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ((mRSubst‘𝑇) “ (𝑅𝑚 𝑉)))
18 fvelima 6207 . . . . . . . 8 ((Fun (mRSubst‘𝑇) ∧ ((mRSubst‘𝑇)‘𝑓) ∈ ((mRSubst‘𝑇) “ (𝑅𝑚 𝑉))) → ∃𝑔 ∈ (𝑅𝑚 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓))
1911, 17, 18syl2anc 692 . . . . . . 7 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ∃𝑔 ∈ (𝑅𝑚 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓))
20 elmapi 7826 . . . . . . . . . . . . 13 (𝑔 ∈ (𝑅𝑚 𝑉) → 𝑔:𝑉𝑅)
2120adantl 482 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → 𝑔:𝑉𝑅)
22 ssid 3605 . . . . . . . . . . . 12 𝑉𝑉
231, 2, 3, 4, 5msubfval 31150 . . . . . . . . . . . 12 ((𝑔:𝑉𝑅𝑉𝑉) → (𝑆𝑔) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩))
2421, 22, 23sylancl 693 . . . . . . . . . . 11 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑆𝑔) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩))
25 fvex 6160 . . . . . . . . . . . . . . . 16 (mEx‘𝑇) ∈ V
2625mptex 6443 . . . . . . . . . . . . . . 15 (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩) ∈ V
27 eqid 2621 . . . . . . . . . . . . . . 15 (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)) = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩))
2826, 27fnmpti 5981 . . . . . . . . . . . . . 14 (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)) Fn (𝑅pm 𝑉)
296fneq1d 5941 . . . . . . . . . . . . . 14 (𝑇 ∈ V → (𝑆 Fn (𝑅pm 𝑉) ↔ (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)) Fn (𝑅pm 𝑉)))
3028, 29mpbiri 248 . . . . . . . . . . . . 13 (𝑇 ∈ V → 𝑆 Fn (𝑅pm 𝑉))
3130adantr 481 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → 𝑆 Fn (𝑅pm 𝑉))
32 mapsspm 7838 . . . . . . . . . . . . 13 (𝑅𝑚 𝑉) ⊆ (𝑅pm 𝑉)
3332a1i 11 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑅𝑚 𝑉) ⊆ (𝑅pm 𝑉))
34 simpr 477 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → 𝑔 ∈ (𝑅𝑚 𝑉))
35 fnfvima 6453 . . . . . . . . . . . 12 ((𝑆 Fn (𝑅pm 𝑉) ∧ (𝑅𝑚 𝑉) ⊆ (𝑅pm 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑆𝑔) ∈ (𝑆 “ (𝑅𝑚 𝑉)))
3631, 33, 34, 35syl3anc 1323 . . . . . . . . . . 11 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑆𝑔) ∈ (𝑆 “ (𝑅𝑚 𝑉)))
3724, 36eqeltrrd 2699 . . . . . . . . . 10 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩) ∈ (𝑆 “ (𝑅𝑚 𝑉)))
3837adantlr 750 . . . . . . . . 9 (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩) ∈ (𝑆 “ (𝑅𝑚 𝑉)))
39 fveq1 6149 . . . . . . . . . . . 12 (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒)) = (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒)))
4039opeq2d 4379 . . . . . . . . . . 11 (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩ = ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)
4140mpteq2dv 4707 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩))
4241eleq1d 2683 . . . . . . . . 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))))
4338, 42syl5ibcom 235 . . . . . . . 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))))
4443rexlimdva 3024 . . . . . . 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))))
4519, 44mpd 15 . . . . . 6 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩) ∈ (𝑆 “ (𝑅𝑚 𝑉)))
4645, 27fmptd 6343 . . . . 5 (𝑇 ∈ V → (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)):(𝑅pm 𝑉)⟶(𝑆 “ (𝑅𝑚 𝑉)))
47 frn 6012 . . . . 5 ((𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)):(𝑅pm 𝑉)⟶(𝑆 “ (𝑅𝑚 𝑉)) → ran (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)) ⊆ (𝑆 “ (𝑅𝑚 𝑉)))
4846, 47syl 17 . . . 4 (𝑇 ∈ V → ran (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)) ⊆ (𝑆 “ (𝑅𝑚 𝑉)))
497, 48eqsstrd 3620 . . 3 (𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅𝑚 𝑉)))
50 fvprc 6144 . . . . . . 7 𝑇 ∈ V → (mSubst‘𝑇) = ∅)
513, 50syl5eq 2667 . . . . . 6 𝑇 ∈ V → 𝑆 = ∅)
5251rneqd 5315 . . . . 5 𝑇 ∈ V → ran 𝑆 = ran ∅)
53 rn0 5339 . . . . 5 ran ∅ = ∅
5452, 53syl6eq 2671 . . . 4 𝑇 ∈ V → ran 𝑆 = ∅)
55 0ss 3946 . . . 4 ∅ ⊆ (𝑆 “ (𝑅𝑚 𝑉))
5654, 55syl6eqss 3636 . . 3 𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅𝑚 𝑉)))
5749, 56pm2.61i 176 . 2 ran 𝑆 ⊆ (𝑆 “ (𝑅𝑚 𝑉))
58 imassrn 5438 . 2 (𝑆 “ (𝑅𝑚 𝑉)) ⊆ ran 𝑆
5957, 58eqssi 3600 1 ran 𝑆 = (𝑆 “ (𝑅𝑚 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3186  wss 3556  c0 3893  cop 4156  cmpt 4675  ran crn 5077  cima 5079  Fun wfun 5843   Fn wfn 5844  wf 5845  cfv 5849  (class class class)co 6607  1st c1st 7114  2nd c2nd 7115  𝑚 cmap 7805  pm cpm 7806  mVRcmvar 31087  mRExcmrex 31092  mExcmex 31093  mRSubstcmrsub 31096  mSubstcmsub 31097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-oadd 7512  df-er 7690  df-map 7807  df-pm 7808  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-card 8712  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-nn 10968  df-2 11026  df-n0 11240  df-z 11325  df-uz 11635  df-fz 12272  df-fzo 12410  df-seq 12745  df-hash 13061  df-word 13241  df-concat 13243  df-s1 13244  df-struct 15786  df-ndx 15787  df-slot 15788  df-base 15789  df-sets 15790  df-ress 15791  df-plusg 15878  df-0g 16026  df-gsum 16027  df-mgm 17166  df-sgrp 17208  df-mnd 17219  df-submnd 17260  df-frmd 17310  df-mrex 31112  df-mrsub 31116  df-msub 31117
This theorem is referenced by:  msubff1o  31183
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