Theorem msubrn 31972

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 “ (𝑅 ↑𝑚 𝑉))

Proof of Theorem msubrn

Dummy variables 𝑒 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		msubff.v	. . . . . 6 ⊢ 𝑉 = (mVR‘𝑇)
2		msubff.r	. . . . . 6 ⊢ 𝑅 = (mREx‘𝑇)
3		msubff.s	. . . . . 6 ⊢ 𝑆 = (mSubst‘𝑇)
4		eqid 2825	. . . . . 6 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
5		eqid 2825	. . . . . 6 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
6	1, 2, 3, 4, 5	msubffval 31966	. . . . 5 ⊢ (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩)))
7	6	rneqd 5585	. . . 4 ⊢ (𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩)))
8	1, 2, 5	mrsubff 31955	. . . . . . . . . 10 ⊢ (𝑇 ∈ V → (mRSubst‘𝑇):(𝑅 ↑pm 𝑉)⟶(𝑅 ↑𝑚 𝑅))
9	8	adantr 474	. . . . . . . . 9 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (mRSubst‘𝑇):(𝑅 ↑pm 𝑉)⟶(𝑅 ↑𝑚 𝑅))
10	9	ffund 6282	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → Fun (mRSubst‘𝑇))
11	8	ffnd 6279	. . . . . . . . . 10 ⊢ (𝑇 ∈ V → (mRSubst‘𝑇) Fn (𝑅 ↑pm 𝑉))
12		fnfvelrn 6605	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇) Fn (𝑅 ↑pm 𝑉) ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
13	11, 12	sylan 577	. . . . . . . . 9 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
14	1, 2, 5	mrsubrn 31956	. . . . . . . . 9 ⊢ ran (mRSubst‘𝑇) = ((mRSubst‘𝑇) “ (𝑅 ↑𝑚 𝑉))
15	13, 14	syl6eleq 2916	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ((mRSubst‘𝑇) “ (𝑅 ↑𝑚 𝑉)))
16		fvelima 6495	. . . . . . . 8 ⊢ ((Fun (mRSubst‘𝑇) ∧ ((mRSubst‘𝑇)‘𝑓) ∈ ((mRSubst‘𝑇) “ (𝑅 ↑𝑚 𝑉))) → ∃𝑔 ∈ (𝑅 ↑𝑚 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓))
17	10, 15, 16	syl2anc 581	. . . . . . 7 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → ∃𝑔 ∈ (𝑅 ↑𝑚 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓))
18		elmapi 8144	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (𝑅 ↑𝑚 𝑉) → 𝑔:𝑉⟶𝑅)
19	18	adantl 475	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑𝑚 𝑉)) → 𝑔:𝑉⟶𝑅)
20		ssid 3848	. . . . . . . . . . . 12 ⊢ 𝑉 ⊆ 𝑉
21	1, 2, 3, 4, 5	msubfval 31967	. . . . . . . . . . . 12 ⊢ ((𝑔:𝑉⟶𝑅 ∧ 𝑉 ⊆ 𝑉) → (𝑆‘𝑔) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘𝑒))⟩))
22	19, 20, 21	sylancl 582	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑𝑚 𝑉)) → (𝑆‘𝑔) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘𝑒))⟩))
23		fvex 6446	. . . . . . . . . . . . . . . 16 ⊢ (mEx‘𝑇) ∈ V
24	23	mptex 6742	. . . . . . . . . . . . . . 15 ⊢ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩) ∈ V
25		eqid 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩)) = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩))
26	24, 25	fnmpti 6255	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩)) Fn (𝑅 ↑pm 𝑉)
27	6	fneq1d 6214	. . . . . . . . . . . . . 14 ⊢ (𝑇 ∈ V → (𝑆 Fn (𝑅 ↑pm 𝑉) ↔ (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩)) Fn (𝑅 ↑pm 𝑉)))
28	26, 27	mpbiri 250	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ V → 𝑆 Fn (𝑅 ↑pm 𝑉))
29	28	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑𝑚 𝑉)) → 𝑆 Fn (𝑅 ↑pm 𝑉))
30		mapsspm 8156	. . . . . . . . . . . . 13 ⊢ (𝑅 ↑𝑚 𝑉) ⊆ (𝑅 ↑pm 𝑉)
31	30	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑𝑚 𝑉)) → (𝑅 ↑𝑚 𝑉) ⊆ (𝑅 ↑pm 𝑉))
32		simpr 479	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑𝑚 𝑉)) → 𝑔 ∈ (𝑅 ↑𝑚 𝑉))
33		fnfvima 6752	. . . . . . . . . . . 12 ⊢ ((𝑆 Fn (𝑅 ↑pm 𝑉) ∧ (𝑅 ↑𝑚 𝑉) ⊆ (𝑅 ↑pm 𝑉) ∧ 𝑔 ∈ (𝑅 ↑𝑚 𝑉)) → (𝑆‘𝑔) ∈ (𝑆 “ (𝑅 ↑𝑚 𝑉)))
34	29, 31, 32, 33	syl3anc 1496	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑𝑚 𝑉)) → (𝑆‘𝑔) ∈ (𝑆 “ (𝑅 ↑𝑚 𝑉)))
35	22, 34	eqeltrrd 2907	. . . . . . . . . 10 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑𝑚 𝑉)) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘𝑒))⟩) ∈ (𝑆 “ (𝑅 ↑𝑚 𝑉)))
36	35	adantlr 708	. . . . . . . . 9 ⊢ (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑔 ∈ (𝑅 ↑𝑚 𝑉)) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘𝑒))⟩) ∈ (𝑆 “ (𝑅 ↑𝑚 𝑉)))
37		fveq1 6432	. . . . . . . . . . . 12 ⊢ (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘𝑒)) = (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒)))
38	37	opeq2d 4630	. . . . . . . . . . 11 ⊢ (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘𝑒))⟩ = ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩)
39	38	mpteq2dv 4968	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘𝑒))⟩) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩))
40	39	eleq1d 2891	. . . . . . . . 9 ⊢ (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → ((𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘𝑒))⟩) ∈ (𝑆 “ (𝑅 ↑𝑚 𝑉)) ↔ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩) ∈ (𝑆 “ (𝑅 ↑𝑚 𝑉))))
41	36, 40	syl5ibcom 237	. . . . . . . 8 ⊢ (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑔 ∈ (𝑅 ↑𝑚 𝑉)) → (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩) ∈ (𝑆 “ (𝑅 ↑𝑚 𝑉))))
42	41	rexlimdva 3240	. . . . . . 7 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (∃𝑔 ∈ (𝑅 ↑𝑚 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩) ∈ (𝑆 “ (𝑅 ↑𝑚 𝑉))))
43	17, 42	mpd 15	. . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩) ∈ (𝑆 “ (𝑅 ↑𝑚 𝑉)))
44	43	fmpttd 6634	. . . . 5 ⊢ (𝑇 ∈ V → (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩)):(𝑅 ↑pm 𝑉)⟶(𝑆 “ (𝑅 ↑𝑚 𝑉)))
45	44	frnd 6285	. . . 4 ⊢ (𝑇 ∈ V → ran (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩)) ⊆ (𝑆 “ (𝑅 ↑𝑚 𝑉)))
46	7, 45	eqsstrd 3864	. . 3 ⊢ (𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑𝑚 𝑉)))
47	3	rnfvprc 6427	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅)
48		0ss 4197	. . . 4 ⊢ ∅ ⊆ (𝑆 “ (𝑅 ↑𝑚 𝑉))
49	47, 48	syl6eqss 3880	. . 3 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑𝑚 𝑉)))
50	46, 49	pm2.61i 177	. 2 ⊢ ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑𝑚 𝑉))
51		imassrn 5718	. 2 ⊢ (𝑆 “ (𝑅 ↑𝑚 𝑉)) ⊆ ran 𝑆
52	50, 51	eqssi 3843	1 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑𝑚 𝑉))

Syntax hints:  ¬ wn 3   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∃wrex 3118  Vcvv 3414   ⊆ wss 3798  ∅c0 4144  ⟨cop 4403   ↦ cmpt 4952  ran crn 5343   “ cima 5345  Fun wfun 6117   Fn wfn 6118  ⟶wf 6119  ‘cfv 6123  (class class class)co 6905  1^st c1st 7426  2^nd c2nd 7427   ↑_𝑚 cmap 8122   ↑_pm cpm 8123  mVRcmvar 31904  mRExcmrex 31909  mExcmex 31910  mRSubstcmrsub 31913  mSubstcmsub 31914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-er 8009  df-map 8124  df-pm 8125  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-card 9078  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-n0 11619  df-z 11705  df-uz 11969  df-fz 12620  df-fzo 12761  df-seq 13096  df-hash 13411  df-word 13575  df-concat 13631  df-s1 13656  df-struct 16224  df-ndx 16225  df-slot 16226  df-base 16228  df-sets 16229  df-ress 16230  df-plusg 16318  df-0g 16455  df-gsum 16456  df-mgm 17595  df-sgrp 17637  df-mnd 17648  df-submnd 17689  df-frmd 17740  df-mrex 31929  df-mrsub 31933  df-msub 31934

This theorem is referenced by:  msubff1o  32000