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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 6906 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇)) | |
3 | mvhfval.v | . . . . . 6 ⊢ 𝑉 = (mVR‘𝑇) | |
4 | 2, 3 | eqtr4di 2792 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉) |
5 | fveq2 6906 | . . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇)) | |
6 | mvhfval.y | . . . . . . . 8 ⊢ 𝑌 = (mType‘𝑇) | |
7 | 5, 6 | eqtr4di 2792 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = 𝑌) |
8 | 7 | fveq1d 6908 | . . . . . 6 ⊢ (𝑡 = 𝑇 → ((mType‘𝑡)‘𝑣) = (𝑌‘𝑣)) |
9 | 8 | opeq1d 4883 | . . . . 5 ⊢ (𝑡 = 𝑇 → 〈((mType‘𝑡)‘𝑣), 〈“𝑣”〉〉 = 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
10 | 4, 9 | mpteq12dv 5238 | . . . 4 ⊢ (𝑡 = 𝑇 → (𝑣 ∈ (mVR‘𝑡) ↦ 〈((mType‘𝑡)‘𝑣), 〈“𝑣”〉〉) = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉)) |
11 | df-mvh 35476 | . . . 4 ⊢ mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ 〈((mType‘𝑡)‘𝑣), 〈“𝑣”〉〉)) | |
12 | 10, 11, 3 | mptfvmpt 7247 | . . 3 ⊢ (𝑇 ∈ V → (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉)) |
13 | mpt0 6710 | . . . . 5 ⊢ (𝑣 ∈ ∅ ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) = ∅ | |
14 | 13 | eqcomi 2743 | . . . 4 ⊢ ∅ = (𝑣 ∈ ∅ ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
15 | fvprc 6898 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mVH‘𝑇) = ∅) | |
16 | fvprc 6898 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mVR‘𝑇) = ∅) | |
17 | 3, 16 | eqtrid 2786 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑉 = ∅) |
18 | 17 | mpteq1d 5242 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) = (𝑣 ∈ ∅ ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉)) |
19 | 14, 15, 18 | 3eqtr4a 2800 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉)) |
20 | 12, 19 | pm2.61i 182 | . 2 ⊢ (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
21 | 1, 20 | eqtri 2762 | 1 ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∅c0 4338 〈cop 4636 ↦ cmpt 5230 ‘cfv 6562 〈“cs1 14629 mVRcmvar 35445 mTypecmty 35446 mVHcmvh 35456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-mvh 35476 |
This theorem is referenced by: mvhval 35518 mvhf 35542 |
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