Theorem mvrsval 35499

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 6899	. . . . 5 ⊢ (𝑋 ∈ (mEx‘𝑇) → 𝑇 ∈ V)
3		mvrsval.e	. . . . 5 ⊢ 𝐸 = (mEx‘𝑇)
4	2, 3	eleq2s 2847	. . . 4 ⊢ (𝑋 ∈ 𝐸 → 𝑇 ∈ V)
5		fveq2 6861	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
6	5, 3	eqtr4di 2783	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
7		fveq2 6861	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
8		mvrsval.v	. . . . . . . 8 ⊢ 𝑉 = (mVR‘𝑇)
9	7, 8	eqtr4di 2783	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
10	9	ineq2d 4186	. . . . . 6 ⊢ (𝑡 = 𝑇 → (ran (2nd ‘𝑒) ∩ (mVR‘𝑡)) = (ran (2nd ‘𝑒) ∩ 𝑉))
11	6, 10	mpteq12dv 5197	. . . . 5 ⊢ (𝑡 = 𝑇 → (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡))) = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉)))
12		df-mvrs 35483	. . . . 5 ⊢ mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡))))
13	11, 12, 3	mptfvmpt 7205	. . . 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 2777	. 2 ⊢ (𝑋 ∈ 𝐸 → 𝑊 = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉)))
16		fveq2 6861	. . . . 5 ⊢ (𝑒 = 𝑋 → (2nd ‘𝑒) = (2nd ‘𝑋))
17	16	rneqd 5905	. . . 4 ⊢ (𝑒 = 𝑋 → ran (2nd ‘𝑒) = ran (2nd ‘𝑋))
18	17	ineq1d 4185	. . 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 6874	. . . . 5 ⊢ (2nd ‘𝑋) ∈ V
22	21	rnex 7889	. . . 4 ⊢ ran (2nd ‘𝑋) ∈ V
23	22	inex1 5275	. . 3 ⊢ (ran (2nd ‘𝑋) ∩ 𝑉) ∈ V
24	23	a1i 11	. 2 ⊢ (𝑋 ∈ 𝐸 → (ran (2nd ‘𝑋) ∩ 𝑉) ∈ V)
25	15, 19, 20, 24	fvmptd 6978	1 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) = (ran (2nd ‘𝑋) ∩ 𝑉))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ∩ cin 3916   ↦ cmpt 5191  ran crn 5642  ‘cfv 6514  2^nd c2nd 7970  mVRcmvar 35455  mExcmex 35461  mVarscmvrs 35463

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  ax-un 7714

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-mvrs 35483

This theorem is referenced by:  mvrsfpw  35500  msubvrs  35554