Theorem msubffval 35545

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 3480	. 2 ⊢ (𝑇 ∈ 𝑊 → 𝑇 ∈ V)
2		msubffval.s	. . 3 ⊢ 𝑆 = (mSubst‘𝑇)
3		fveq2 6876	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇))
4		msubffval.r	. . . . . . 7 ⊢ 𝑅 = (mREx‘𝑇)
5	3, 4	eqtr4di 2788	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅)
6		fveq2 6876	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
7		msubffval.v	. . . . . . 7 ⊢ 𝑉 = (mVR‘𝑇)
8	6, 7	eqtr4di 2788	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
9	5, 8	oveq12d 7423	. . . . 5 ⊢ (𝑡 = 𝑇 → ((mREx‘𝑡) ↑pm (mVR‘𝑡)) = (𝑅 ↑pm 𝑉))
10		fveq2 6876	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
11		msubffval.e	. . . . . . 7 ⊢ 𝐸 = (mEx‘𝑇)
12	10, 11	eqtr4di 2788	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
13		fveq2 6876	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (mRSubst‘𝑡) = (mRSubst‘𝑇))
14		msubffval.o	. . . . . . . . . 10 ⊢ 𝑂 = (mRSubst‘𝑇)
15	13, 14	eqtr4di 2788	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mRSubst‘𝑡) = 𝑂)
16	15	fveq1d 6878	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → ((mRSubst‘𝑡)‘𝑓) = (𝑂‘𝑓))
17	16	fveq1d 6878	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (((mRSubst‘𝑡)‘𝑓)‘(2nd ‘𝑒)) = ((𝑂‘𝑓)‘(2nd ‘𝑒)))
18	17	opeq2d 4856	. . . . . 6 ⊢ (𝑡 = 𝑇 → ⟨(1st ‘𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd ‘𝑒))⟩ = ⟨(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))⟩)
19	12, 18	mpteq12dv 5207	. . . . 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 5207	. . . 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 35513	. . . 4 ⊢ mSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mEx‘𝑡) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd ‘𝑒))⟩)))
22		ovex 7438	. . . . 5 ⊢ (𝑅 ↑pm 𝑉) ∈ V
23	22	mptex 7215	. . . 4 ⊢ (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))⟩)) ∈ V
24	20, 21, 23	fvmpt 6986	. . 3 ⊢ (𝑇 ∈ V → (mSubst‘𝑇) = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))⟩)))
25	2, 24	eqtrid 2782	. 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 1540   ∈ wcel 2108  Vcvv 3459  ⟨cop 4607   ↦ cmpt 5201  ‘cfv 6531  (class class class)co 7405  1^st c1st 7986  2^nd c2nd 7987   ↑_pm cpm 8841  mVRcmvar 35483  mRExcmrex 35488  mExcmex 35489  mRSubstcmrsub 35492  mSubstcmsub 35493

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-msub 35513

This theorem is referenced by:  msubfval  35546  elmsubrn  35550  msubrn  35551  msubff  35552