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Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mvhfval | Structured version Visualization version GIF version |
Description: Value of the function mapping variables to their corresponding variable expressions. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mvhfval.v | ⊢ 𝑉 = (mVR‘𝑇) |
mvhfval.y | ⊢ 𝑌 = (mType‘𝑇) |
mvhfval.h | ⊢ 𝐻 = (mVH‘𝑇) |
Ref | Expression |
---|---|
mvhfval | ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mvhfval.h | . 2 ⊢ 𝐻 = (mVH‘𝑇) | |
2 | fveq2 6332 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇)) | |
3 | mvhfval.v | . . . . . 6 ⊢ 𝑉 = (mVR‘𝑇) | |
4 | 2, 3 | syl6eqr 2823 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉) |
5 | fveq2 6332 | . . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇)) | |
6 | mvhfval.y | . . . . . . . 8 ⊢ 𝑌 = (mType‘𝑇) | |
7 | 5, 6 | syl6eqr 2823 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = 𝑌) |
8 | 7 | fveq1d 6334 | . . . . . 6 ⊢ (𝑡 = 𝑇 → ((mType‘𝑡)‘𝑣) = (𝑌‘𝑣)) |
9 | 8 | opeq1d 4545 | . . . . 5 ⊢ (𝑡 = 𝑇 → 〈((mType‘𝑡)‘𝑣), 〈“𝑣”〉〉 = 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
10 | 4, 9 | mpteq12dv 4867 | . . . 4 ⊢ (𝑡 = 𝑇 → (𝑣 ∈ (mVR‘𝑡) ↦ 〈((mType‘𝑡)‘𝑣), 〈“𝑣”〉〉) = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉)) |
11 | df-mvh 31727 | . . . 4 ⊢ mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ 〈((mType‘𝑡)‘𝑣), 〈“𝑣”〉〉)) | |
12 | fvex 6342 | . . . . . 6 ⊢ (mVR‘𝑇) ∈ V | |
13 | 3, 12 | eqeltri 2846 | . . . . 5 ⊢ 𝑉 ∈ V |
14 | 13 | mptex 6630 | . . . 4 ⊢ (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) ∈ V |
15 | 10, 11, 14 | fvmpt 6424 | . . 3 ⊢ (𝑇 ∈ V → (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉)) |
16 | mpt0 6161 | . . . . 5 ⊢ (𝑣 ∈ ∅ ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) = ∅ | |
17 | 16 | eqcomi 2780 | . . . 4 ⊢ ∅ = (𝑣 ∈ ∅ ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
18 | fvprc 6326 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mVH‘𝑇) = ∅) | |
19 | fvprc 6326 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mVR‘𝑇) = ∅) | |
20 | 3, 19 | syl5eq 2817 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑉 = ∅) |
21 | 20 | mpteq1d 4872 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) = (𝑣 ∈ ∅ ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉)) |
22 | 17, 18, 21 | 3eqtr4a 2831 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉)) |
23 | 15, 22 | pm2.61i 176 | . 2 ⊢ (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
24 | 1, 23 | eqtri 2793 | 1 ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1631 ∈ wcel 2145 Vcvv 3351 ∅c0 4063 〈cop 4322 ↦ cmpt 4863 ‘cfv 6031 〈“cs1 13490 mVRcmvar 31696 mTypecmty 31697 mVHcmvh 31707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-mvh 31727 |
This theorem is referenced by: mvhval 31769 mvhf 31793 |
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