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Theorem elmsubrn 35515

Description: Characterization of substitution in terms of raw substitution, without reference to the generating functions. (Contributed by Mario Carneiro, 18-Jul-2016.)

**Hypotheses**
Ref	Expression
elmsubrn.e	⊢ 𝐸 = (mEx‘𝑇)
elmsubrn.o	⊢ 𝑂 = (mRSubst‘𝑇)
elmsubrn.s	⊢ 𝑆 = (mSubst‘𝑇)

**Assertion**
Ref	Expression
elmsubrn	⊢ ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩))

Distinct variable groups: 𝑒,𝑓,𝐸 𝑒,𝑂,𝑓 𝑇,𝑒 Allowed substitution hints: 𝑆(𝑒,𝑓) 𝑇(𝑓)

Proof of Theorem elmsubrn Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. . . . . 6 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
2		eqid 2729	. . . . . 6 ⊢ (mREx‘𝑇) = (mREx‘𝑇)
3		elmsubrn.s	. . . . . 6 ⊢ 𝑆 = (mSubst‘𝑇)
4		elmsubrn.e	. . . . . 6 ⊢ 𝐸 = (mEx‘𝑇)
5		elmsubrn.o	. . . . . 6 ⊢ 𝑂 = (mRSubst‘𝑇)
6	1, 2, 3, 4, 5	msubffval 35510	. . . . 5 ⊢ (𝑇 ∈ V → 𝑆 = (𝑔 ∈ ((mREx‘𝑇) ↑_pm (mVR‘𝑇)) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), ((𝑂‘𝑔)‘(2^nd ‘𝑒))⟩)))
7	1, 2, 5	mrsubff 35499	. . . . . . . 8 ⊢ (𝑇 ∈ V → 𝑂:((mREx‘𝑇) ↑_pm (mVR‘𝑇))⟶((mREx‘𝑇) ↑_m (mREx‘𝑇)))
8	7	ffnd 6689	. . . . . . 7 ⊢ (𝑇 ∈ V → 𝑂 Fn ((mREx‘𝑇) ↑_pm (mVR‘𝑇)))
9		fnfvelrn 7052	. . . . . . 7 ⊢ ((𝑂 Fn ((mREx‘𝑇) ↑_pm (mVR‘𝑇)) ∧ 𝑔 ∈ ((mREx‘𝑇) ↑_pm (mVR‘𝑇))) → (𝑂‘𝑔) ∈ ran 𝑂)
10	8, 9	sylan 580	. . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ ((mREx‘𝑇) ↑_pm (mVR‘𝑇))) → (𝑂‘𝑔) ∈ ran 𝑂)
11	7	feqmptd 6929	. . . . . 6 ⊢ (𝑇 ∈ V → 𝑂 = (𝑔 ∈ ((mREx‘𝑇) ↑_pm (mVR‘𝑇)) ↦ (𝑂‘𝑔)))
12		eqidd 2730	. . . . . 6 ⊢ (𝑇 ∈ V → (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)) = (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)))
13		fveq1 6857	. . . . . . . 8 ⊢ (𝑓 = (𝑂‘𝑔) → (𝑓‘(2^nd ‘𝑒)) = ((𝑂‘𝑔)‘(2^nd ‘𝑒)))
14	13	opeq2d 4844	. . . . . . 7 ⊢ (𝑓 = (𝑂‘𝑔) → ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩ = ⟨(1^st ‘𝑒), ((𝑂‘𝑔)‘(2^nd ‘𝑒))⟩)
15	14	mpteq2dv 5201	. . . . . 6 ⊢ (𝑓 = (𝑂‘𝑔) → (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩) = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), ((𝑂‘𝑔)‘(2^nd ‘𝑒))⟩))
16	10, 11, 12, 15	fmptco 7101	. . . . 5 ⊢ (𝑇 ∈ V → ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)) ∘ 𝑂) = (𝑔 ∈ ((mREx‘𝑇) ↑_pm (mVR‘𝑇)) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), ((𝑂‘𝑔)‘(2^nd ‘𝑒))⟩)))
17	6, 16	eqtr4d 2767	. . . 4 ⊢ (𝑇 ∈ V → 𝑆 = ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)) ∘ 𝑂))
18	17	rneqd 5902	. . 3 ⊢ (𝑇 ∈ V → ran 𝑆 = ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)) ∘ 𝑂))
19		rnco 6225	. . . 4 ⊢ ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)) ∘ 𝑂) = ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)) ↾ ran 𝑂)
20		ssid 3969	. . . . . 6 ⊢ ran 𝑂 ⊆ ran 𝑂
21		resmpt 6008	. . . . . 6 ⊢ (ran 𝑂 ⊆ ran 𝑂 → ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)) ↾ ran 𝑂) = (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)))
22	20, 21	ax-mp 5	. . . . 5 ⊢ ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)) ↾ ran 𝑂) = (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩))
23	22	rneqi 5901	. . . 4 ⊢ ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)) ↾ ran 𝑂) = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩))
24	19, 23	eqtri 2752	. . 3 ⊢ ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)) ∘ 𝑂) = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩))
25	18, 24	eqtrdi 2780	. 2 ⊢ (𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)))
26		mpt0 6660	. . . . 5 ⊢ (𝑓 ∈ ∅ ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)) = ∅
27	26	eqcomi 2738	. . . 4 ⊢ ∅ = (𝑓 ∈ ∅ ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩))
28		fvprc 6850	. . . . 5 ⊢ (¬ 𝑇 ∈ V → (mSubst‘𝑇) = ∅)
29	3, 28	eqtrid 2776	. . . 4 ⊢ (¬ 𝑇 ∈ V → 𝑆 = ∅)
30	5	rnfvprc 6852	. . . . 5 ⊢ (¬ 𝑇 ∈ V → ran 𝑂 = ∅)
31	30	mpteq1d 5197	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)) = (𝑓 ∈ ∅ ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)))
32	27, 29, 31	3eqtr4a 2790	. . 3 ⊢ (¬ 𝑇 ∈ V → 𝑆 = (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)))
33	32	rneqd 5902	. 2 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)))
34	25, 33	pm2.61i 182	1 ⊢ ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2109 Vcvv 3447 ⊆ wss 3914 ∅c0 4296 ⟨cop 4595 ↦ cmpt 5188 ran crn 5639 ↾ cres 5640 ∘ ccom 5642 Fn wfn 6506 ‘cfv 6511 (class class class)co 7387 1^st c1st 7966 2^nd c2nd 7967 ↑_m cmap 8799 ↑_pm cpm 8800 mVRcmvar 35448 mRExcmrex 35453 mExcmex 35454 mRSubstcmrsub 35457 mSubstcmsub 35458

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 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-pm 8802 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469 df-fzo 13616 df-seq 13967 df-hash 14296 df-word 14479 df-concat 14536 df-s1 14561 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-0g 17404 df-gsum 17405 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-frmd 18776 df-mrex 35473 df-mrsub 35477 df-msub 35478

This theorem is referenced by: msubco 35518 msubvrs 35547

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