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Theorem elmsubrn 35471

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 35466	. . . . 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 35455	. . . . . . . 8 ⊢ (𝑇 ∈ V → 𝑂:((mREx‘𝑇) ↑_pm (mVR‘𝑇))⟶((mREx‘𝑇) ↑_m (mREx‘𝑇)))
8	7	ffnd 6703	. . . . . . 7 ⊢ (𝑇 ∈ V → 𝑂 Fn ((mREx‘𝑇) ↑_pm (mVR‘𝑇)))
9		fnfvelrn 7066	. . . . . . 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 6943	. . . . . 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 6871	. . . . . . . 8 ⊢ (𝑓 = (𝑂‘𝑔) → (𝑓‘(2^nd ‘𝑒)) = ((𝑂‘𝑔)‘(2^nd ‘𝑒)))
14	13	opeq2d 4853	. . . . . . 7 ⊢ (𝑓 = (𝑂‘𝑔) → ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩ = ⟨(1^st ‘𝑒), ((𝑂‘𝑔)‘(2^nd ‘𝑒))⟩)
15	14	mpteq2dv 5212	. . . . . 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 7115	. . . . 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 5915	. . 3 ⊢ (𝑇 ∈ V → ran 𝑆 = ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)) ∘ 𝑂))
19		rnco 6238	. . . 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 3979	. . . . . 6 ⊢ ran 𝑂 ⊆ ran 𝑂
21		resmpt 6021	. . . . . 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 5914	. . . 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 6676	. . . . 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 6864	. . . . 5 ⊢ (¬ 𝑇 ∈ V → (mSubst‘𝑇) = ∅)
29	3, 28	eqtrid 2781	. . . 4 ⊢ (¬ 𝑇 ∈ V → 𝑆 = ∅)
30	5	rnfvprc 6866	. . . . 5 ⊢ (¬ 𝑇 ∈ V → ran 𝑂 = ∅)
31	30	mpteq1d 5207	. . . 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 5915	. 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 1539 ∈ wcel 2107 Vcvv 3457 ⊆ wss 3924 ∅c0 4306 ⟨cop 4605 ↦ cmpt 5198 ran crn 5652 ↾ cres 5653 ∘ ccom 5655 Fn wfn 6522 ‘cfv 6527 (class class class)co 7399 1^st c1st 7980 2^nd c2nd 7981 ↑_m cmap 8834 ↑_pm cpm 8835 mVRcmvar 35404 mRExcmrex 35409 mExcmex 35410 mRSubstcmrsub 35413 mSubstcmsub 35414

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5246 ax-sep 5263 ax-nul 5273 ax-pow 5332 ax-pr 5399 ax-un 7723 ax-cnex 11177 ax-resscn 11178 ax-1cn 11179 ax-icn 11180 ax-addcl 11181 ax-addrcl 11182 ax-mulcl 11183 ax-mulrcl 11184 ax-mulcom 11185 ax-addass 11186 ax-mulass 11187 ax-distr 11188 ax-i2m1 11189 ax-1ne0 11190 ax-1rid 11191 ax-rnegex 11192 ax-rrecex 11193 ax-cnre 11194 ax-pre-lttri 11195 ax-pre-lttrn 11196 ax-pre-ltadd 11197 ax-pre-mulgt0 11198

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3357 df-reu 3358 df-rab 3414 df-v 3459 df-sbc 3764 df-csb 3873 df-dif 3927 df-un 3929 df-in 3931 df-ss 3941 df-pss 3944 df-nul 4307 df-if 4499 df-pw 4575 df-sn 4600 df-pr 4602 df-op 4606 df-uni 4881 df-int 4920 df-iun 4966 df-br 5117 df-opab 5179 df-mpt 5199 df-tr 5227 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6287 df-ord 6352 df-on 6353 df-lim 6354 df-suc 6355 df-iota 6480 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7356 df-ov 7402 df-oprab 7403 df-mpo 7404 df-om 7856 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8379 df-rdg 8418 df-1o 8474 df-er 8713 df-map 8836 df-pm 8837 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-card 9945 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11460 df-neg 11461 df-nn 12233 df-2 12295 df-n0 12494 df-z 12581 df-uz 12845 df-fz 13514 df-fzo 13661 df-seq 14009 df-hash 14337 df-word 14520 df-concat 14576 df-s1 14601 df-struct 17151 df-sets 17168 df-slot 17186 df-ndx 17198 df-base 17214 df-ress 17237 df-plusg 17269 df-0g 17440 df-gsum 17441 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18747 df-frmd 18812 df-mrex 35429 df-mrsub 35433 df-msub 35434

This theorem is referenced by: msubco 35474 msubvrs 35503

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