Theorem mvrsval 35819

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 6898	. . . . 5 ⊢ (𝑋 ∈ (mEx‘𝑇) → 𝑇 ∈ V)
3		mvrsval.e	. . . . 5 ⊢ 𝐸 = (mEx‘𝑇)
4	2, 3	eleq2s 2879	. . . 4 ⊢ (𝑋 ∈ 𝐸 → 𝑇 ∈ V)
5		fveq2 6863	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
6	5, 3	eqtr4di 2814	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
7		fveq2 6863	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
8		mvrsval.v	. . . . . . . 8 ⊢ 𝑉 = (mVR‘𝑇)
9	7, 8	eqtr4di 2814	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
10	9	ineq2d 4172	. . . . . 6 ⊢ (𝑡 = 𝑇 → (ran (2nd ‘𝑒) ∩ (mVR‘𝑡)) = (ran (2nd ‘𝑒) ∩ 𝑉))
11	6, 10	mpteq12dv 5186	. . . . 5 ⊢ (𝑡 = 𝑇 → (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡))) = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉)))
12		df-mvrs 35803	. . . . 5 ⊢ mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡))))
13	11, 12, 3	mptfvmpt 7208	. . . 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 2808	. 2 ⊢ (𝑋 ∈ 𝐸 → 𝑊 = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉)))
16		fveq2 6863	. . . . 5 ⊢ (𝑒 = 𝑋 → (2nd ‘𝑒) = (2nd ‘𝑋))
17	16	rneqd 5912	. . . 4 ⊢ (𝑒 = 𝑋 → ran (2nd ‘𝑒) = ran (2nd ‘𝑋))
18	17	ineq1d 4171	. . 3 ⊢ (𝑒 = 𝑋 → (ran (2nd ‘𝑒) ∩ 𝑉) = (ran (2nd ‘𝑋) ∩ 𝑉))
19	18	adantl 485	. 2 ⊢ ((𝑋 ∈ 𝐸 ∧ 𝑒 = 𝑋) → (ran (2nd ‘𝑒) ∩ 𝑉) = (ran (2nd ‘𝑋) ∩ 𝑉))
20		id 22	. 2 ⊢ (𝑋 ∈ 𝐸 → 𝑋 ∈ 𝐸)
21		fvex 6876	. . . . 5 ⊢ (2nd ‘𝑋) ∈ V
22	21	rnex 7887	. . . 4 ⊢ ran (2nd ‘𝑋) ∈ V
23	22	inex1 5272	. . 3 ⊢ (ran (2nd ‘𝑋) ∩ 𝑉) ∈ V
24	23	a1i 11	. 2 ⊢ (𝑋 ∈ 𝐸 → (ran (2nd ‘𝑋) ∩ 𝑉) ∈ V)
25	15, 19, 20, 24	fvmptd 6979	1 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) = (ran (2nd ‘𝑋) ∩ 𝑉))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ∩ cin 3903   ↦ cmpt 5180  ran crn 5646  ‘cfv 6517  2^nd c2nd 7965  mVRcmvar 35775  mExcmex 35781  mVarscmvrs 35783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-mvrs 35803

This theorem is referenced by:  mvrsfpw  35820  msubvrs  35874