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 6703 | . . . . 5 ⊢ (𝑋 ∈ (mEx‘𝑇) → 𝑇 ∈ V) | |
3 | mvrsval.e | . . . . 5 ⊢ 𝐸 = (mEx‘𝑇) | |
4 | 2, 3 | eleq2s 2931 | . . . 4 ⊢ (𝑋 ∈ 𝐸 → 𝑇 ∈ V) |
5 | fveq2 6670 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇)) | |
6 | 5, 3 | syl6eqr 2874 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸) |
7 | fveq2 6670 | . . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇)) | |
8 | mvrsval.v | . . . . . . . 8 ⊢ 𝑉 = (mVR‘𝑇) | |
9 | 7, 8 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉) |
10 | 9 | ineq2d 4189 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (ran (2nd ‘𝑒) ∩ (mVR‘𝑡)) = (ran (2nd ‘𝑒) ∩ 𝑉)) |
11 | 6, 10 | mpteq12dv 5151 | . . . . 5 ⊢ (𝑡 = 𝑇 → (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡))) = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉))) |
12 | df-mvrs 32736 | . . . . 5 ⊢ mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡)))) | |
13 | 11, 12, 3 | mptfvmpt 6990 | . . . 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 2868 | . 2 ⊢ (𝑋 ∈ 𝐸 → 𝑊 = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉))) |
16 | fveq2 6670 | . . . . 5 ⊢ (𝑒 = 𝑋 → (2nd ‘𝑒) = (2nd ‘𝑋)) | |
17 | 16 | rneqd 5808 | . . . 4 ⊢ (𝑒 = 𝑋 → ran (2nd ‘𝑒) = ran (2nd ‘𝑋)) |
18 | 17 | ineq1d 4188 | . . 3 ⊢ (𝑒 = 𝑋 → (ran (2nd ‘𝑒) ∩ 𝑉) = (ran (2nd ‘𝑋) ∩ 𝑉)) |
19 | 18 | adantl 484 | . 2 ⊢ ((𝑋 ∈ 𝐸 ∧ 𝑒 = 𝑋) → (ran (2nd ‘𝑒) ∩ 𝑉) = (ran (2nd ‘𝑋) ∩ 𝑉)) |
20 | id 22 | . 2 ⊢ (𝑋 ∈ 𝐸 → 𝑋 ∈ 𝐸) | |
21 | fvex 6683 | . . . . 5 ⊢ (2nd ‘𝑋) ∈ V | |
22 | 21 | rnex 7617 | . . . 4 ⊢ ran (2nd ‘𝑋) ∈ V |
23 | 22 | inex1 5221 | . . 3 ⊢ (ran (2nd ‘𝑋) ∩ 𝑉) ∈ V |
24 | 23 | a1i 11 | . 2 ⊢ (𝑋 ∈ 𝐸 → (ran (2nd ‘𝑋) ∩ 𝑉) ∈ V) |
25 | 15, 19, 20, 24 | fvmptd 6775 | 1 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) = (ran (2nd ‘𝑋) ∩ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∩ cin 3935 ↦ cmpt 5146 ran crn 5556 ‘cfv 6355 2nd c2nd 7688 mVRcmvar 32708 mExcmex 32714 mVarscmvrs 32716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-mvrs 32736 |
This theorem is referenced by: mvrsfpw 32753 msubvrs 32807 |
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