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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 6944 | . . . . 5 ⊢ (𝑋 ∈ (mEx‘𝑇) → 𝑇 ∈ V) | |
| 3 | mvrsval.e | . . . . 5 ⊢ 𝐸 = (mEx‘𝑇) | |
| 4 | 2, 3 | eleq2s 2859 | . . . 4 ⊢ (𝑋 ∈ 𝐸 → 𝑇 ∈ V) |
| 5 | fveq2 6906 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇)) | |
| 6 | 5, 3 | eqtr4di 2795 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸) |
| 7 | fveq2 6906 | . . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇)) | |
| 8 | mvrsval.v | . . . . . . . 8 ⊢ 𝑉 = (mVR‘𝑇) | |
| 9 | 7, 8 | eqtr4di 2795 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉) |
| 10 | 9 | ineq2d 4220 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (ran (2nd ‘𝑒) ∩ (mVR‘𝑡)) = (ran (2nd ‘𝑒) ∩ 𝑉)) |
| 11 | 6, 10 | mpteq12dv 5233 | . . . . 5 ⊢ (𝑡 = 𝑇 → (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡))) = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉))) |
| 12 | df-mvrs 35494 | . . . . 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 2789 | . 2 ⊢ (𝑋 ∈ 𝐸 → 𝑊 = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉))) |
| 16 | fveq2 6906 | . . . . 5 ⊢ (𝑒 = 𝑋 → (2nd ‘𝑒) = (2nd ‘𝑋)) | |
| 17 | 16 | rneqd 5949 | . . . 4 ⊢ (𝑒 = 𝑋 → ran (2nd ‘𝑒) = ran (2nd ‘𝑋)) |
| 18 | 17 | ineq1d 4219 | . . 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 6919 | . . . . 5 ⊢ (2nd ‘𝑋) ∈ V | |
| 22 | 21 | rnex 7932 | . . . 4 ⊢ ran (2nd ‘𝑋) ∈ V |
| 23 | 22 | inex1 5317 | . . 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 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ↦ cmpt 5225 ran crn 5686 ‘cfv 6561 2nd c2nd 8013 mVRcmvar 35466 mExcmex 35472 mVarscmvrs 35474 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-mvrs 35494 |
| This theorem is referenced by: mvrsfpw 35511 msubvrs 35565 |
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