msubrn - Mathbox for Mario Carneiro

	Mathbox for Mario Carneiro		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > msubrn		Structured version Visualization version GIF version

Theorem msubrn 35523

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 2730	. . . . . 6 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
5		eqid 2730	. . . . . 6 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
6	1, 2, 3, 4, 5	msubffval 35517	. . . . 5 ⊢ (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)))
7	6	rneqd 5905	. . . 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 35506	. . . . . . . . . 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 6695	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → Fun (mRSubst‘𝑇))
11	8	ffnd 6692	. . . . . . . . . 10 ⊢ (𝑇 ∈ V → (mRSubst‘𝑇) Fn (𝑅 ↑_pm 𝑉))
12		fnfvelrn 7055	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇) Fn (𝑅 ↑_pm 𝑉) ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
13	11, 12	sylan 580	. . . . . . . . 9 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
14	1, 2, 5	mrsubrn 35507	. . . . . . . . 9 ⊢ ran (mRSubst‘𝑇) = ((mRSubst‘𝑇) “ (𝑅 ↑_m 𝑉))
15	13, 14	eleqtrdi 2839	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ((mRSubst‘𝑇) “ (𝑅 ↑_m 𝑉)))
16		fvelima 6929	. . . . . . . 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 584	. . . . . . 7 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ∃𝑔 ∈ (𝑅 ↑_m 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓))
18		elmapi 8825	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (𝑅 ↑_m 𝑉) → 𝑔:𝑉⟶𝑅)
19	18	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → 𝑔:𝑉⟶𝑅)
20		ssid 3972	. . . . . . . . . . . 12 ⊢ 𝑉 ⊆ 𝑉
21	1, 2, 3, 4, 5	msubfval 35518	. . . . . . . . . . . 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 586	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → (𝑆‘𝑔) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒))⟩))
23		fvex 6874	. . . . . . . . . . . . . . . 16 ⊢ (mEx‘𝑇) ∈ V
24	23	mptex 7200	. . . . . . . . . . . . . . 15 ⊢ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩) ∈ V
25		eqid 2730	. . . . . . . . . . . . . . 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 6664	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)) Fn (𝑅 ↑_pm 𝑉)
27	6	fneq1d 6614	. . . . . . . . . . . . . 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 8852	. . . . . . . . . . . . 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 7210	. . . . . . . . . . . 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 1373	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → (𝑆‘𝑔) ∈ (𝑆 “ (𝑅 ↑_m 𝑉)))
35	22, 34	eqeltrrd 2830	. . . . . . . . . 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 715	. . . . . . . . 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 6860	. . . . . . . . . . . 12 ⊢ (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒)) = (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒)))
38	37	opeq2d 4847	. . . . . . . . . . 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 5204	. . . . . . . . . 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 2814	. . . . . . . . 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 3135	. . . . . . 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 7090	. . . . 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 6699	. . . 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 3984	. . 3 ⊢ (𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑_m 𝑉)))
47	3	rnfvprc 6855	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅)
48		0ss 4366	. . . 4 ⊢ ∅ ⊆ (𝑆 “ (𝑅 ↑_m 𝑉))
49	47, 48	eqsstrdi 3994	. . 3 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑_m 𝑉)))
50	46, 49	pm2.61i 182	. 2 ⊢ ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑_m 𝑉))
51		imassrn 6045	. 2 ⊢ (𝑆 “ (𝑅 ↑_m 𝑉)) ⊆ ran 𝑆
52	50, 51	eqssi 3966	1 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑_m 𝑉))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∃wrex 3054 Vcvv 3450 ⊆ wss 3917 ∅c0 4299 ⟨cop 4598 ↦ cmpt 5191 ran crn 5642 “ cima 5644 Fun wfun 6508 Fn wfn 6509 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 1^st c1st 7969 2^nd c2nd 7970 ↑_m cmap 8802 ↑_pm cpm 8803 mVRcmvar 35455 mRExcmrex 35460 mExcmex 35461 mRSubstcmrsub 35464 mSubstcmsub 35465

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-map 8804 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-fzo 13623 df-seq 13974 df-hash 14303 df-word 14486 df-concat 14543 df-s1 14568 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-0g 17411 df-gsum 17412 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-frmd 18783 df-mrex 35480 df-mrsub 35484 df-msub 35485

This theorem is referenced by: msubff1o 35551

W3C validator