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 6774 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇)) | |
3 | mvhfval.v | . . . . . 6 ⊢ 𝑉 = (mVR‘𝑇) | |
4 | 2, 3 | eqtr4di 2796 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉) |
5 | fveq2 6774 | . . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇)) | |
6 | mvhfval.y | . . . . . . . 8 ⊢ 𝑌 = (mType‘𝑇) | |
7 | 5, 6 | eqtr4di 2796 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = 𝑌) |
8 | 7 | fveq1d 6776 | . . . . . 6 ⊢ (𝑡 = 𝑇 → ((mType‘𝑡)‘𝑣) = (𝑌‘𝑣)) |
9 | 8 | opeq1d 4810 | . . . . 5 ⊢ (𝑡 = 𝑇 → 〈((mType‘𝑡)‘𝑣), 〈“𝑣”〉〉 = 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
10 | 4, 9 | mpteq12dv 5165 | . . . 4 ⊢ (𝑡 = 𝑇 → (𝑣 ∈ (mVR‘𝑡) ↦ 〈((mType‘𝑡)‘𝑣), 〈“𝑣”〉〉) = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉)) |
11 | df-mvh 33454 | . . . 4 ⊢ mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ 〈((mType‘𝑡)‘𝑣), 〈“𝑣”〉〉)) | |
12 | 10, 11, 3 | mptfvmpt 7104 | . . 3 ⊢ (𝑇 ∈ V → (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉)) |
13 | mpt0 6575 | . . . . 5 ⊢ (𝑣 ∈ ∅ ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) = ∅ | |
14 | 13 | eqcomi 2747 | . . . 4 ⊢ ∅ = (𝑣 ∈ ∅ ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
15 | fvprc 6766 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mVH‘𝑇) = ∅) | |
16 | fvprc 6766 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mVR‘𝑇) = ∅) | |
17 | 3, 16 | eqtrid 2790 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑉 = ∅) |
18 | 17 | mpteq1d 5169 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) = (𝑣 ∈ ∅ ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉)) |
19 | 14, 15, 18 | 3eqtr4a 2804 | . . 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 2766 | 1 ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ∅c0 4256 〈cop 4567 ↦ cmpt 5157 ‘cfv 6433 〈“cs1 14300 mVRcmvar 33423 mTypecmty 33424 mVHcmvh 33434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-mvh 33454 |
This theorem is referenced by: mvhval 33496 mvhf 33520 |
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