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Theorem elmsubrn 35671

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 2734	. . . . . 6 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
2		eqid 2734	. . . . . 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 35666	. . . . 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 35655	. . . . . . . 8 ⊢ (𝑇 ∈ V → 𝑂:((mREx‘𝑇) ↑_pm (mVR‘𝑇))⟶((mREx‘𝑇) ↑_m (mREx‘𝑇)))
8	7	ffnd 6661	. . . . . . 7 ⊢ (𝑇 ∈ V → 𝑂 Fn ((mREx‘𝑇) ↑_pm (mVR‘𝑇)))
9		fnfvelrn 7023	. . . . . . 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 6900	. . . . . 6 ⊢ (𝑇 ∈ V → 𝑂 = (𝑔 ∈ ((mREx‘𝑇) ↑_pm (mVR‘𝑇)) ↦ (𝑂‘𝑔)))
12		eqidd 2735	. . . . . 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 6831	. . . . . . . 8 ⊢ (𝑓 = (𝑂‘𝑔) → (𝑓‘(2^nd ‘𝑒)) = ((𝑂‘𝑔)‘(2^nd ‘𝑒)))
14	13	opeq2d 4834	. . . . . . 7 ⊢ (𝑓 = (𝑂‘𝑔) → ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩ = ⟨(1^st ‘𝑒), ((𝑂‘𝑔)‘(2^nd ‘𝑒))⟩)
15	14	mpteq2dv 5190	. . . . . 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 7072	. . . . 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 2772	. . . 4 ⊢ (𝑇 ∈ V → 𝑆 = ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)) ∘ 𝑂))
18	17	rneqd 5885	. . 3 ⊢ (𝑇 ∈ V → ran 𝑆 = ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)) ∘ 𝑂))
19		rnco 6208	. . . 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 3954	. . . . . 6 ⊢ ran 𝑂 ⊆ ran 𝑂
21		resmpt 5994	. . . . . 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 5884	. . . 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 2757	. . 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 2785	. 2 ⊢ (𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)))
26		mpt0 6632	. . . . 5 ⊢ (𝑓 ∈ ∅ ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)) = ∅
27	26	eqcomi 2743	. . . 4 ⊢ ∅ = (𝑓 ∈ ∅ ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩))
28		fvprc 6824	. . . . 5 ⊢ (¬ 𝑇 ∈ V → (mSubst‘𝑇) = ∅)
29	3, 28	eqtrid 2781	. . . 4 ⊢ (¬ 𝑇 ∈ V → 𝑆 = ∅)
30	5	rnfvprc 6826	. . . . 5 ⊢ (¬ 𝑇 ∈ V → ran 𝑂 = ∅)
31	30	mpteq1d 5186	. . . 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 2795	. . 3 ⊢ (¬ 𝑇 ∈ V → 𝑆 = (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)))
33	32	rneqd 5885	. 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 1541 ∈ wcel 2113 Vcvv 3438 ⊆ wss 3899 ∅c0 4283 ⟨cop 4584 ↦ cmpt 5177 ran crn 5623 ↾ cres 5624 ∘ ccom 5626 Fn wfn 6485 ‘cfv 6490 (class class class)co 7356 1^st c1st 7929 2^nd c2nd 7930 ↑_m cmap 8761 ↑_pm cpm 8762 mVRcmvar 35604 mRExcmrex 35609 mExcmex 35610 mRSubstcmrsub 35613 mSubstcmsub 35614

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8763 df-pm 8764 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-n0 12400 df-z 12487 df-uz 12750 df-fz 13422 df-fzo 13569 df-seq 13923 df-hash 14252 df-word 14435 df-concat 14492 df-s1 14518 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-0g 17359 df-gsum 17360 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-frmd 18772 df-mrex 35629 df-mrsub 35633 df-msub 35634

This theorem is referenced by: msubco 35674 msubvrs 35703

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