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Theorem msubrn 31552
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 2651 . . . . . 6 (mEx‘𝑇) = (mEx‘𝑇)
5 eqid 2651 . . . . . 6 (mRSubst‘𝑇) = (mRSubst‘𝑇)
61, 2, 3, 4, 5msubffval 31546 . . . . 5 (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)))
76rneqd 5385 . . . 4 (𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)))
81, 2, 5mrsubff 31535 . . . . . . . . . 10 (𝑇 ∈ V → (mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅))
98adantr 480 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → (mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅))
10 ffun 6086 . . . . . . . . 9 ((mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅) → Fun (mRSubst‘𝑇))
119, 10syl 17 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → Fun (mRSubst‘𝑇))
12 ffn 6083 . . . . . . . . . . 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 6396 . . . . . . . . . 10 (((mRSubst‘𝑇) Fn (𝑅pm 𝑉) ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
1513, 14sylan 487 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
161, 2, 5mrsubrn 31536 . . . . . . . . 9 ran (mRSubst‘𝑇) = ((mRSubst‘𝑇) “ (𝑅𝑚 𝑉))
1715, 16syl6eleq 2740 . . . . . . . 8 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ((mRSubst‘𝑇) “ (𝑅𝑚 𝑉)))
18 fvelima 6287 . . . . . . . 8 ((Fun (mRSubst‘𝑇) ∧ ((mRSubst‘𝑇)‘𝑓) ∈ ((mRSubst‘𝑇) “ (𝑅𝑚 𝑉))) → ∃𝑔 ∈ (𝑅𝑚 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓))
1911, 17, 18syl2anc 694 . . . . . . 7 ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) → ∃𝑔 ∈ (𝑅𝑚 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓))
20 elmapi 7921 . . . . . . . . . . . . 13 (𝑔 ∈ (𝑅𝑚 𝑉) → 𝑔:𝑉𝑅)
2120adantl 481 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → 𝑔:𝑉𝑅)
22 ssid 3657 . . . . . . . . . . . 12 𝑉𝑉
231, 2, 3, 4, 5msubfval 31547 . . . . . . . . . . . 12 ((𝑔:𝑉𝑅𝑉𝑉) → (𝑆𝑔) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩))
2421, 22, 23sylancl 695 . . . . . . . . . . 11 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑆𝑔) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩))
25 fvex 6239 . . . . . . . . . . . . . . . 16 (mEx‘𝑇) ∈ V
2625mptex 6527 . . . . . . . . . . . . . . 15 (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩) ∈ V
27 eqid 2651 . . . . . . . . . . . . . . 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 6060 . . . . . . . . . . . . . 14 (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)) Fn (𝑅pm 𝑉)
296fneq1d 6019 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → 𝑆 Fn (𝑅pm 𝑉))
32 mapsspm 7933 . . . . . . . . . . . . 13 (𝑅𝑚 𝑉) ⊆ (𝑅pm 𝑉)
3332a1i 11 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑅𝑚 𝑉) ⊆ (𝑅pm 𝑉))
34 simpr 476 . . . . . . . . . . . 12 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → 𝑔 ∈ (𝑅𝑚 𝑉))
35 fnfvima 6536 . . . . . . . . . . . 12 ((𝑆 Fn (𝑅pm 𝑉) ∧ (𝑅𝑚 𝑉) ⊆ (𝑅pm 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑆𝑔) ∈ (𝑆 “ (𝑅𝑚 𝑉)))
3631, 33, 34, 35syl3anc 1366 . . . . . . . . . . 11 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑆𝑔) ∈ (𝑆 “ (𝑅𝑚 𝑉)))
3724, 36eqeltrrd 2731 . . . . . . . . . 10 ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩) ∈ (𝑆 “ (𝑅𝑚 𝑉)))
3837adantlr 751 . . . . . . . . 9 (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩) ∈ (𝑆 “ (𝑅𝑚 𝑉)))
39 fveq1 6228 . . . . . . . . . . . 12 (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒)) = (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒)))
4039opeq2d 4440 . . . . . . . . . . 11 (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩ = ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)
4140mpteq2dv 4778 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd𝑒))⟩) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩))
4241eleq1d 2715 . . . . . . . . 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 3060 . . . . . . 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 6425 . . . . 5 (𝑇 ∈ V → (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)):(𝑅pm 𝑉)⟶(𝑆 “ (𝑅𝑚 𝑉)))
47 frn 6091 . . . . 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 3672 . . 3 (𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅𝑚 𝑉)))
50 fvprc 6223 . . . . . . 7 𝑇 ∈ V → (mSubst‘𝑇) = ∅)
513, 50syl5eq 2697 . . . . . 6 𝑇 ∈ V → 𝑆 = ∅)
5251rneqd 5385 . . . . 5 𝑇 ∈ V → ran 𝑆 = ran ∅)
53 rn0 5409 . . . . 5 ran ∅ = ∅
5452, 53syl6eq 2701 . . . 4 𝑇 ∈ V → ran 𝑆 = ∅)
55 0ss 4005 . . . 4 ∅ ⊆ (𝑆 “ (𝑅𝑚 𝑉))
5654, 55syl6eqss 3688 . . 3 𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅𝑚 𝑉)))
5749, 56pm2.61i 176 . 2 ran 𝑆 ⊆ (𝑆 “ (𝑅𝑚 𝑉))
58 imassrn 5512 . 2 (𝑆 “ (𝑅𝑚 𝑉)) ⊆ ran 𝑆
5957, 58eqssi 3652 1 ran 𝑆 = (𝑆 “ (𝑅𝑚 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1523  wcel 2030  wrex 2942  Vcvv 3231  wss 3607  c0 3948  cop 4216  cmpt 4762  ran crn 5144  cima 5146  Fun wfun 5920   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  𝑚 cmap 7899  pm cpm 7900  mVRcmvar 31484  mRExcmrex 31489  mExcmex 31490  mRSubstcmrsub 31493  mSubstcmsub 31494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-word 13331  df-concat 13333  df-s1 13334  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-0g 16149  df-gsum 16150  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-frmd 17433  df-mrex 31509  df-mrsub 31513  df-msub 31514
This theorem is referenced by:  msubff1o  31580
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