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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 6945 | . . . . 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 6907 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇)) | |
6 | 5, 3 | eqtr4di 2793 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸) |
7 | fveq2 6907 | . . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇)) | |
8 | mvrsval.v | . . . . . . . 8 ⊢ 𝑉 = (mVR‘𝑇) | |
9 | 7, 8 | eqtr4di 2793 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉) |
10 | 9 | ineq2d 4228 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (ran (2nd ‘𝑒) ∩ (mVR‘𝑡)) = (ran (2nd ‘𝑒) ∩ 𝑉)) |
11 | 6, 10 | mpteq12dv 5239 | . . . . 5 ⊢ (𝑡 = 𝑇 → (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡))) = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉))) |
12 | df-mvrs 35474 | . . . . 5 ⊢ mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡)))) | |
13 | 11, 12, 3 | mptfvmpt 7248 | . . . 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 2787 | . 2 ⊢ (𝑋 ∈ 𝐸 → 𝑊 = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉))) |
16 | fveq2 6907 | . . . . 5 ⊢ (𝑒 = 𝑋 → (2nd ‘𝑒) = (2nd ‘𝑋)) | |
17 | 16 | rneqd 5952 | . . . 4 ⊢ (𝑒 = 𝑋 → ran (2nd ‘𝑒) = ran (2nd ‘𝑋)) |
18 | 17 | ineq1d 4227 | . . 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 6920 | . . . . 5 ⊢ (2nd ‘𝑋) ∈ V | |
22 | 21 | rnex 7933 | . . . 4 ⊢ ran (2nd ‘𝑋) ∈ V |
23 | 22 | inex1 5323 | . . 3 ⊢ (ran (2nd ‘𝑋) ∩ 𝑉) ∈ V |
24 | 23 | a1i 11 | . 2 ⊢ (𝑋 ∈ 𝐸 → (ran (2nd ‘𝑋) ∩ 𝑉) ∈ V) |
25 | 15, 19, 20, 24 | fvmptd 7023 | 1 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) = (ran (2nd ‘𝑋) ∩ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∩ cin 3962 ↦ cmpt 5231 ran crn 5690 ‘cfv 6563 2nd c2nd 8012 mVRcmvar 35446 mExcmex 35452 mVarscmvrs 35454 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-mvrs 35474 |
This theorem is referenced by: mvrsfpw 35491 msubvrs 35545 |
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