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Theorem elmsubrn 35710

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 2736	. . . . . 6 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
2		eqid 2736	. . . . . 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 35705	. . . . 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 35694	. . . . . . . 8 ⊢ (𝑇 ∈ V → 𝑂:((mREx‘𝑇) ↑_pm (mVR‘𝑇))⟶((mREx‘𝑇) ↑_m (mREx‘𝑇)))
8	7	ffnd 6669	. . . . . . 7 ⊢ (𝑇 ∈ V → 𝑂 Fn ((mREx‘𝑇) ↑_pm (mVR‘𝑇)))
9		fnfvelrn 7032	. . . . . . 7 ⊢ ((𝑂 Fn ((mREx‘𝑇) ↑_pm (mVR‘𝑇)) ∧ 𝑔 ∈ ((mREx‘𝑇) ↑_pm (mVR‘𝑇))) → (𝑂‘𝑔) ∈ ran 𝑂)
10	8, 9	sylan 581	. . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ ((mREx‘𝑇) ↑_pm (mVR‘𝑇))) → (𝑂‘𝑔) ∈ ran 𝑂)
11	7	feqmptd 6908	. . . . . 6 ⊢ (𝑇 ∈ V → 𝑂 = (𝑔 ∈ ((mREx‘𝑇) ↑_pm (mVR‘𝑇)) ↦ (𝑂‘𝑔)))
12		eqidd 2737	. . . . . 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 6839	. . . . . . . 8 ⊢ (𝑓 = (𝑂‘𝑔) → (𝑓‘(2^nd ‘𝑒)) = ((𝑂‘𝑔)‘(2^nd ‘𝑒)))
14	13	opeq2d 4823	. . . . . . 7 ⊢ (𝑓 = (𝑂‘𝑔) → ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩ = ⟨(1^st ‘𝑒), ((𝑂‘𝑔)‘(2^nd ‘𝑒))⟩)
15	14	mpteq2dv 5179	. . . . . 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 7082	. . . . 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 2774	. . . 4 ⊢ (𝑇 ∈ V → 𝑆 = ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)) ∘ 𝑂))
18	17	rneqd 5893	. . 3 ⊢ (𝑇 ∈ V → ran 𝑆 = ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)) ∘ 𝑂))
19		rnco 6216	. . . 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 3944	. . . . . 6 ⊢ ran 𝑂 ⊆ ran 𝑂
21		resmpt 6002	. . . . . 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 5892	. . . 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 2759	. . 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 2787	. 2 ⊢ (𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)))
26		mpt0 6640	. . . . 5 ⊢ (𝑓 ∈ ∅ ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)) = ∅
27	26	eqcomi 2745	. . . 4 ⊢ ∅ = (𝑓 ∈ ∅ ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩))
28		fvprc 6832	. . . . 5 ⊢ (¬ 𝑇 ∈ V → (mSubst‘𝑇) = ∅)
29	3, 28	eqtrid 2783	. . . 4 ⊢ (¬ 𝑇 ∈ V → 𝑆 = ∅)
30	5	rnfvprc 6834	. . . . 5 ⊢ (¬ 𝑇 ∈ V → ran 𝑂 = ∅)
31	30	mpteq1d 5175	. . . 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 2797	. . 3 ⊢ (¬ 𝑇 ∈ V → 𝑆 = (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)))
33	32	rneqd 5893	. 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 1542 ∈ wcel 2114 Vcvv 3429 ⊆ wss 3889 ∅c0 4273 ⟨cop 4573 ↦ cmpt 5166 ran crn 5632 ↾ cres 5633 ∘ ccom 5635 Fn wfn 6493 ‘cfv 6498 (class class class)co 7367 1^st c1st 7940 2^nd c2nd 7941 ↑_m cmap 8773 ↑_pm cpm 8774 mVRcmvar 35643 mRExcmrex 35648 mExcmex 35649 mRSubstcmrsub 35652 mSubstcmsub 35653

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 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-map 8775 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-fzo 13609 df-seq 13964 df-hash 14293 df-word 14476 df-concat 14533 df-s1 14559 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 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-frmd 18817 df-mrex 35668 df-mrsub 35672 df-msub 35673

This theorem is referenced by: msubco 35713 msubvrs 35742

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