msubrn - Mathbox for Mario Carneiro

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Theorem msubrn 35745

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 2737	. . . . . 6 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
5		eqid 2737	. . . . . 6 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
6	1, 2, 3, 4, 5	msubffval 35739	. . . . 5 ⊢ (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)))
7	6	rneqd 5895	. . . 4 ⊢ (𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)))
8	1, 2, 5	mrsubff 35728	. . . . . . . . . 10 ⊢ (𝑇 ∈ V → (mRSubst‘𝑇):(𝑅 ↑_pm 𝑉)⟶(𝑅 ↑_m 𝑅))
9	8	adantr 480	. . . . . . . . 9 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → (mRSubst‘𝑇):(𝑅 ↑_pm 𝑉)⟶(𝑅 ↑_m 𝑅))
10	9	ffund 6674	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → Fun (mRSubst‘𝑇))
11	8	ffnd 6671	. . . . . . . . . 10 ⊢ (𝑇 ∈ V → (mRSubst‘𝑇) Fn (𝑅 ↑_pm 𝑉))
12		fnfvelrn 7034	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇) Fn (𝑅 ↑_pm 𝑉) ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
13	11, 12	sylan 581	. . . . . . . . 9 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
14	1, 2, 5	mrsubrn 35729	. . . . . . . . 9 ⊢ ran (mRSubst‘𝑇) = ((mRSubst‘𝑇) “ (𝑅 ↑_m 𝑉))
15	13, 14	eleqtrdi 2847	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ((mRSubst‘𝑇) “ (𝑅 ↑_m 𝑉)))
16		fvelima 6907	. . . . . . . 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 585	. . . . . . 7 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ∃𝑔 ∈ (𝑅 ↑_m 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓))
18		elmapi 8798	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (𝑅 ↑_m 𝑉) → 𝑔:𝑉⟶𝑅)
19	18	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → 𝑔:𝑉⟶𝑅)
20		ssid 3958	. . . . . . . . . . . 12 ⊢ 𝑉 ⊆ 𝑉
21	1, 2, 3, 4, 5	msubfval 35740	. . . . . . . . . . . 12 ⊢ ((𝑔:𝑉⟶𝑅 ∧ 𝑉 ⊆ 𝑉) → (𝑆‘𝑔) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒))⟩))
22	19, 20, 21	sylancl 587	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → (𝑆‘𝑔) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒))⟩))
23		fvex 6855	. . . . . . . . . . . . . . . 16 ⊢ (mEx‘𝑇) ∈ V
24	23	mptex 7179	. . . . . . . . . . . . . . 15 ⊢ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩) ∈ V
25		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)) = (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩))
26	24, 25	fnmpti 6643	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)) Fn (𝑅 ↑_pm 𝑉)
27	6	fneq1d 6593	. . . . . . . . . . . . . 14 ⊢ (𝑇 ∈ V → (𝑆 Fn (𝑅 ↑_pm 𝑉) ↔ (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)) Fn (𝑅 ↑_pm 𝑉)))
28	26, 27	mpbiri 258	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ V → 𝑆 Fn (𝑅 ↑_pm 𝑉))
29	28	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → 𝑆 Fn (𝑅 ↑_pm 𝑉))
30		mapsspm 8826	. . . . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → 𝑔 ∈ (𝑅 ↑_m 𝑉))
33		fnfvima 7189	. . . . . . . . . . . 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 1374	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → (𝑆‘𝑔) ∈ (𝑆 “ (𝑅 ↑_m 𝑉)))
35	22, 34	eqeltrrd 2838	. . . . . . . . . 10 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒))⟩) ∈ (𝑆 “ (𝑅 ↑_m 𝑉)))
36	35	adantlr 716	. . . . . . . . 9 ⊢ (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒))⟩) ∈ (𝑆 “ (𝑅 ↑_m 𝑉)))
37		fveq1 6841	. . . . . . . . . . . 12 ⊢ (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒)) = (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒)))
38	37	opeq2d 4838	. . . . . . . . . . 11 ⊢ (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒))⟩ = ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)
39	38	mpteq2dv 5194	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒))⟩) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩))
40	39	eleq1d 2822	. . . . . . . . 9 ⊢ (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → ((𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒))⟩) ∈ (𝑆 “ (𝑅 ↑_m 𝑉)) ↔ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩) ∈ (𝑆 “ (𝑅 ↑_m 𝑉))))
41	36, 40	syl5ibcom 245	. . . . . . . 8 ⊢ (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩) ∈ (𝑆 “ (𝑅 ↑_m 𝑉))))
42	41	rexlimdva 3139	. . . . . . 7 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → (∃𝑔 ∈ (𝑅 ↑_m 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩) ∈ (𝑆 “ (𝑅 ↑_m 𝑉))))
43	17, 42	mpd 15	. . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩) ∈ (𝑆 “ (𝑅 ↑_m 𝑉)))
44	43	fmpttd 7069	. . . . 5 ⊢ (𝑇 ∈ V → (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)):(𝑅 ↑_pm 𝑉)⟶(𝑆 “ (𝑅 ↑_m 𝑉)))
45	44	frnd 6678	. . . 4 ⊢ (𝑇 ∈ V → ran (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)) ⊆ (𝑆 “ (𝑅 ↑_m 𝑉)))
46	7, 45	eqsstrd 3970	. . 3 ⊢ (𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑_m 𝑉)))
47	3	rnfvprc 6836	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅)
48		0ss 4354	. . . 4 ⊢ ∅ ⊆ (𝑆 “ (𝑅 ↑_m 𝑉))
49	47, 48	eqsstrdi 3980	. . 3 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑_m 𝑉)))
50	46, 49	pm2.61i 182	. 2 ⊢ ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑_m 𝑉))
51		imassrn 6038	. 2 ⊢ (𝑆 “ (𝑅 ↑_m 𝑉)) ⊆ ran 𝑆
52	50, 51	eqssi 3952	1 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑_m 𝑉))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∃wrex 3062 Vcvv 3442 ⊆ wss 3903 ∅c0 4287 ⟨cop 4588 ↦ cmpt 5181 ran crn 5633 “ cima 5635 Fun wfun 6494 Fn wfn 6495 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 1^st c1st 7941 2^nd c2nd 7942 ↑_m cmap 8775 ↑_pm cpm 8776 mVRcmvar 35677 mRExcmrex 35682 mExcmex 35683 mRSubstcmrsub 35686 mSubstcmsub 35687

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-pm 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-fzo 13583 df-seq 13937 df-hash 14266 df-word 14449 df-concat 14506 df-s1 14532 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-0g 17373 df-gsum 17374 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-frmd 18786 df-mrex 35702 df-mrsub 35706 df-msub 35707

This theorem is referenced by: msubff1o 35773

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