Theorem mvrsval 33367

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 6789	. . . . 5 ⊢ (𝑋 ∈ (mEx‘𝑇) → 𝑇 ∈ V)
3		mvrsval.e	. . . . 5 ⊢ 𝐸 = (mEx‘𝑇)
4	2, 3	eleq2s 2857	. . . 4 ⊢ (𝑋 ∈ 𝐸 → 𝑇 ∈ V)
5		fveq2 6756	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
6	5, 3	eqtr4di 2797	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
7		fveq2 6756	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
8		mvrsval.v	. . . . . . . 8 ⊢ 𝑉 = (mVR‘𝑇)
9	7, 8	eqtr4di 2797	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
10	9	ineq2d 4143	. . . . . 6 ⊢ (𝑡 = 𝑇 → (ran (2nd ‘𝑒) ∩ (mVR‘𝑡)) = (ran (2nd ‘𝑒) ∩ 𝑉))
11	6, 10	mpteq12dv 5161	. . . . 5 ⊢ (𝑡 = 𝑇 → (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡))) = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉)))
12		df-mvrs 33351	. . . . 5 ⊢ mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡))))
13	11, 12, 3	mptfvmpt 7086	. . . 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	syl5eq 2791	. 2 ⊢ (𝑋 ∈ 𝐸 → 𝑊 = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉)))
16		fveq2 6756	. . . . 5 ⊢ (𝑒 = 𝑋 → (2nd ‘𝑒) = (2nd ‘𝑋))
17	16	rneqd 5836	. . . 4 ⊢ (𝑒 = 𝑋 → ran (2nd ‘𝑒) = ran (2nd ‘𝑋))
18	17	ineq1d 4142	. . 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 6769	. . . . 5 ⊢ (2nd ‘𝑋) ∈ V
22	21	rnex 7733	. . . 4 ⊢ ran (2nd ‘𝑋) ∈ V
23	22	inex1 5236	. . 3 ⊢ (ran (2nd ‘𝑋) ∩ 𝑉) ∈ V
24	23	a1i 11	. 2 ⊢ (𝑋 ∈ 𝐸 → (ran (2nd ‘𝑋) ∩ 𝑉) ∈ V)
25	15, 19, 20, 24	fvmptd 6864	1 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) = (ran (2nd ‘𝑋) ∩ 𝑉))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∩ cin 3882   ↦ cmpt 5153  ran crn 5581  ‘cfv 6418  2^nd c2nd 7803  mVRcmvar 33323  mExcmex 33329  mVarscmvrs 33331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-mvrs 33351

This theorem is referenced by:  mvrsfpw  33368  msubvrs  33422