Theorem mvrsval 35537

Description: The set of variables in an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mvrsval.v	⊢ 𝑉 = (mVR‘𝑇)
mvrsval.e	⊢ 𝐸 = (mEx‘𝑇)
mvrsval.w	⊢ 𝑊 = (mVars‘𝑇)

Assertion

Ref	Expression
mvrsval	⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) = (ran (2nd ‘𝑋) ∩ 𝑉))

Proof of Theorem mvrsval

Dummy variables 𝑡 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mvrsval.w	. . 3 ⊢ 𝑊 = (mVars‘𝑇)
2		elfvex 6857	. . . . 5 ⊢ (𝑋 ∈ (mEx‘𝑇) → 𝑇 ∈ V)
3		mvrsval.e	. . . . 5 ⊢ 𝐸 = (mEx‘𝑇)
4	2, 3	eleq2s 2849	. . . 4 ⊢ (𝑋 ∈ 𝐸 → 𝑇 ∈ V)
5		fveq2 6822	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
6	5, 3	eqtr4di 2784	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
7		fveq2 6822	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
8		mvrsval.v	. . . . . . . 8 ⊢ 𝑉 = (mVR‘𝑇)
9	7, 8	eqtr4di 2784	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
10	9	ineq2d 4170	. . . . . 6 ⊢ (𝑡 = 𝑇 → (ran (2nd ‘𝑒) ∩ (mVR‘𝑡)) = (ran (2nd ‘𝑒) ∩ 𝑉))
11	6, 10	mpteq12dv 5178	. . . . 5 ⊢ (𝑡 = 𝑇 → (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡))) = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉)))
12		df-mvrs 35521	. . . . 5 ⊢ mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡))))
13	11, 12, 3	mptfvmpt 7162	. . . 4 ⊢ (𝑇 ∈ V → (mVars‘𝑇) = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉)))
14	4, 13	syl 17	. . 3 ⊢ (𝑋 ∈ 𝐸 → (mVars‘𝑇) = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉)))
15	1, 14	eqtrid 2778	. 2 ⊢ (𝑋 ∈ 𝐸 → 𝑊 = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉)))
16		fveq2 6822	. . . . 5 ⊢ (𝑒 = 𝑋 → (2nd ‘𝑒) = (2nd ‘𝑋))
17	16	rneqd 5878	. . . 4 ⊢ (𝑒 = 𝑋 → ran (2nd ‘𝑒) = ran (2nd ‘𝑋))
18	17	ineq1d 4169	. . 3 ⊢ (𝑒 = 𝑋 → (ran (2nd ‘𝑒) ∩ 𝑉) = (ran (2nd ‘𝑋) ∩ 𝑉))
19	18	adantl 481	. 2 ⊢ ((𝑋 ∈ 𝐸 ∧ 𝑒 = 𝑋) → (ran (2nd ‘𝑒) ∩ 𝑉) = (ran (2nd ‘𝑋) ∩ 𝑉))
20		id 22	. 2 ⊢ (𝑋 ∈ 𝐸 → 𝑋 ∈ 𝐸)
21		fvex 6835	. . . . 5 ⊢ (2nd ‘𝑋) ∈ V
22	21	rnex 7840	. . . 4 ⊢ ran (2nd ‘𝑋) ∈ V
23	22	inex1 5255	. . 3 ⊢ (ran (2nd ‘𝑋) ∩ 𝑉) ∈ V
24	23	a1i 11	. 2 ⊢ (𝑋 ∈ 𝐸 → (ran (2nd ‘𝑋) ∩ 𝑉) ∈ V)
25	15, 19, 20, 24	fvmptd 6936	1 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) = (ran (2nd ‘𝑋) ∩ 𝑉))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∩ cin 3901   ↦ cmpt 5172  ran crn 5617  ‘cfv 6481  2^nd c2nd 7920  mVRcmvar 35493  mExcmex 35499  mVarscmvrs 35501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-mvrs 35521

This theorem is referenced by:  mvrsfpw  35538  msubvrs  35592