Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mvrsval | Structured version Visualization version GIF version |
Description: The set of variables in an expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mvrsval.v | ⊢ 𝑉 = (mVR‘𝑇) |
mvrsval.e | ⊢ 𝐸 = (mEx‘𝑇) |
mvrsval.w | ⊢ 𝑊 = (mVars‘𝑇) |
Ref | Expression |
---|---|
mvrsval | ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) = (ran (2nd ‘𝑋) ∩ 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mvrsval.w | . . 3 ⊢ 𝑊 = (mVars‘𝑇) | |
2 | elfvex 6696 | . . . . 5 ⊢ (𝑋 ∈ (mEx‘𝑇) → 𝑇 ∈ V) | |
3 | mvrsval.e | . . . . 5 ⊢ 𝐸 = (mEx‘𝑇) | |
4 | 2, 3 | eleq2s 2928 | . . . 4 ⊢ (𝑋 ∈ 𝐸 → 𝑇 ∈ V) |
5 | fveq2 6663 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇)) | |
6 | 5, 3 | syl6eqr 2871 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸) |
7 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇)) | |
8 | mvrsval.v | . . . . . . . 8 ⊢ 𝑉 = (mVR‘𝑇) | |
9 | 7, 8 | syl6eqr 2871 | . . . . . . 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 5142 | . . . . 5 ⊢ (𝑡 = 𝑇 → (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡))) = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉))) |
12 | df-mvrs 32633 | . . . . 5 ⊢ mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡)))) | |
13 | 11, 12, 3 | mptfvmpt 6981 | . . . 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 2865 | . 2 ⊢ (𝑋 ∈ 𝐸 → 𝑊 = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉))) |
16 | fveq2 6663 | . . . . 5 ⊢ (𝑒 = 𝑋 → (2nd ‘𝑒) = (2nd ‘𝑋)) | |
17 | 16 | rneqd 5801 | . . . 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 482 | . 2 ⊢ ((𝑋 ∈ 𝐸 ∧ 𝑒 = 𝑋) → (ran (2nd ‘𝑒) ∩ 𝑉) = (ran (2nd ‘𝑋) ∩ 𝑉)) |
20 | id 22 | . 2 ⊢ (𝑋 ∈ 𝐸 → 𝑋 ∈ 𝐸) | |
21 | fvex 6676 | . . . . 5 ⊢ (2nd ‘𝑋) ∈ V | |
22 | 21 | rnex 7606 | . . . 4 ⊢ ran (2nd ‘𝑋) ∈ V |
23 | 22 | inex1 5212 | . . 3 ⊢ (ran (2nd ‘𝑋) ∩ 𝑉) ∈ V |
24 | 23 | a1i 11 | . 2 ⊢ (𝑋 ∈ 𝐸 → (ran (2nd ‘𝑋) ∩ 𝑉) ∈ V) |
25 | 15, 19, 20, 24 | fvmptd 6767 | 1 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) = (ran (2nd ‘𝑋) ∩ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∩ cin 3932 ↦ cmpt 5137 ran crn 5549 ‘cfv 6348 2nd c2nd 7677 mVRcmvar 32605 mExcmex 32611 mVarscmvrs 32613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-mvrs 32633 |
This theorem is referenced by: mvrsfpw 32650 msubvrs 32704 |
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