Theorem msubffval 35517

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 3471	. 2 ⊢ (𝑇 ∈ 𝑊 → 𝑇 ∈ V)
2		msubffval.s	. . 3 ⊢ 𝑆 = (mSubst‘𝑇)
3		fveq2 6861	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇))
4		msubffval.r	. . . . . . 7 ⊢ 𝑅 = (mREx‘𝑇)
5	3, 4	eqtr4di 2783	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅)
6		fveq2 6861	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
7		msubffval.v	. . . . . . 7 ⊢ 𝑉 = (mVR‘𝑇)
8	6, 7	eqtr4di 2783	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
9	5, 8	oveq12d 7408	. . . . 5 ⊢ (𝑡 = 𝑇 → ((mREx‘𝑡) ↑pm (mVR‘𝑡)) = (𝑅 ↑pm 𝑉))
10		fveq2 6861	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
11		msubffval.e	. . . . . . 7 ⊢ 𝐸 = (mEx‘𝑇)
12	10, 11	eqtr4di 2783	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
13		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (mRSubst‘𝑡) = (mRSubst‘𝑇))
14		msubffval.o	. . . . . . . . . 10 ⊢ 𝑂 = (mRSubst‘𝑇)
15	13, 14	eqtr4di 2783	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mRSubst‘𝑡) = 𝑂)
16	15	fveq1d 6863	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → ((mRSubst‘𝑡)‘𝑓) = (𝑂‘𝑓))
17	16	fveq1d 6863	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (((mRSubst‘𝑡)‘𝑓)‘(2nd ‘𝑒)) = ((𝑂‘𝑓)‘(2nd ‘𝑒)))
18	17	opeq2d 4847	. . . . . 6 ⊢ (𝑡 = 𝑇 → ⟨(1st ‘𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd ‘𝑒))⟩ = ⟨(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))⟩)
19	12, 18	mpteq12dv 5197	. . . . 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 5197	. . . 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 35485	. . . 4 ⊢ mSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mEx‘𝑡) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd ‘𝑒))⟩)))
22		ovex 7423	. . . . 5 ⊢ (𝑅 ↑pm 𝑉) ∈ V
23	22	mptex 7200	. . . 4 ⊢ (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))⟩)) ∈ V
24	20, 21, 23	fvmpt 6971	. . 3 ⊢ (𝑇 ∈ V → (mSubst‘𝑇) = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))⟩)))
25	2, 24	eqtrid 2777	. 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 2109  Vcvv 3450  ⟨cop 4598   ↦ cmpt 5191  ‘cfv 6514  (class class class)co 7390  1^st c1st 7969  2^nd c2nd 7970   ↑_pm cpm 8803  mVRcmvar 35455  mRExcmrex 35460  mExcmex 35461  mRSubstcmrsub 35464  mSubstcmsub 35465

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-msub 35485

This theorem is referenced by:  msubfval  35518  elmsubrn  35522  msubrn  35523  msubff  35524