Theorem msubffval 32019

Description: A substitution applied to an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
msubffval.v	⊢ 𝑉 = (mVR‘𝑇)
msubffval.r	⊢ 𝑅 = (mREx‘𝑇)
msubffval.s	⊢ 𝑆 = (mSubst‘𝑇)
msubffval.e	⊢ 𝐸 = (mEx‘𝑇)
msubffval.o	⊢ 𝑂 = (mRSubst‘𝑇)

Assertion

Ref	Expression
msubffval	⊢ (𝑇 ∈ 𝑊 → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))⟩)))

Distinct variable groups:   𝑒,𝑓,𝐸   𝑒,𝑂,𝑓   𝑅,𝑒,𝑓   𝑇,𝑒,𝑓   𝑒,𝑉,𝑓

Allowed substitution hints:   𝑆(𝑒,𝑓)   𝑊(𝑒,𝑓)

Proof of Theorem msubffval

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3414	. 2 ⊢ (𝑇 ∈ 𝑊 → 𝑇 ∈ V)
2		msubffval.s	. . 3 ⊢ 𝑆 = (mSubst‘𝑇)
3		fveq2 6446	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇))
4		msubffval.r	. . . . . . 7 ⊢ 𝑅 = (mREx‘𝑇)
5	3, 4	syl6eqr 2832	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅)
6		fveq2 6446	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
7		msubffval.v	. . . . . . 7 ⊢ 𝑉 = (mVR‘𝑇)
8	6, 7	syl6eqr 2832	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
9	5, 8	oveq12d 6940	. . . . 5 ⊢ (𝑡 = 𝑇 → ((mREx‘𝑡) ↑pm (mVR‘𝑡)) = (𝑅 ↑pm 𝑉))
10		fveq2 6446	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
11		msubffval.e	. . . . . . 7 ⊢ 𝐸 = (mEx‘𝑇)
12	10, 11	syl6eqr 2832	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
13		fveq2 6446	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (mRSubst‘𝑡) = (mRSubst‘𝑇))
14		msubffval.o	. . . . . . . . . 10 ⊢ 𝑂 = (mRSubst‘𝑇)
15	13, 14	syl6eqr 2832	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mRSubst‘𝑡) = 𝑂)
16	15	fveq1d 6448	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → ((mRSubst‘𝑡)‘𝑓) = (𝑂‘𝑓))
17	16	fveq1d 6448	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (((mRSubst‘𝑡)‘𝑓)‘(2nd ‘𝑒)) = ((𝑂‘𝑓)‘(2nd ‘𝑒)))
18	17	opeq2d 4643	. . . . . 6 ⊢ (𝑡 = 𝑇 → ⟨(1st ‘𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd ‘𝑒))⟩ = ⟨(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))⟩)
19	12, 18	mpteq12dv 4969	. . . . 5 ⊢ (𝑡 = 𝑇 → (𝑒 ∈ (mEx‘𝑡) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd ‘𝑒))⟩) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))⟩))
20	9, 19	mpteq12dv 4969	. . . 4 ⊢ (𝑡 = 𝑇 → (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mEx‘𝑡) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd ‘𝑒))⟩)) = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))⟩)))
21		df-msub 31987	. . . 4 ⊢ mSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mEx‘𝑡) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd ‘𝑒))⟩)))
22		ovex 6954	. . . . 5 ⊢ (𝑅 ↑pm 𝑉) ∈ V
23	22	mptex 6758	. . . 4 ⊢ (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))⟩)) ∈ V
24	20, 21, 23	fvmpt 6542	. . 3 ⊢ (𝑇 ∈ V → (mSubst‘𝑇) = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))⟩)))
25	2, 24	syl5eq 2826	. 2 ⊢ (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))⟩)))
26	1, 25	syl 17	1 ⊢ (𝑇 ∈ 𝑊 → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))⟩)))

Syntax hints:   → wi 4   = wceq 1601   ∈ wcel 2107  Vcvv 3398  ⟨cop 4404   ↦ cmpt 4965  ‘cfv 6135  (class class class)co 6922  1^st c1st 7443  2^nd c2nd 7444   ↑_pm cpm 8141  mVRcmvar 31957  mRExcmrex 31962  mExcmex 31963  mRSubstcmrsub 31966  mSubstcmsub 31967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-msub 31987

This theorem is referenced by:  msubfval  32020  elmsubrn  32024  msubrn  32025  msubff  32026