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 6664 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇)) | |
3 | mvhfval.v | . . . . . 6 ⊢ 𝑉 = (mVR‘𝑇) | |
4 | 2, 3 | syl6eqr 2874 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉) |
5 | fveq2 6664 | . . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇)) | |
6 | mvhfval.y | . . . . . . . 8 ⊢ 𝑌 = (mType‘𝑇) | |
7 | 5, 6 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = 𝑌) |
8 | 7 | fveq1d 6666 | . . . . . 6 ⊢ (𝑡 = 𝑇 → ((mType‘𝑡)‘𝑣) = (𝑌‘𝑣)) |
9 | 8 | opeq1d 4802 | . . . . 5 ⊢ (𝑡 = 𝑇 → 〈((mType‘𝑡)‘𝑣), 〈“𝑣”〉〉 = 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
10 | 4, 9 | mpteq12dv 5143 | . . . 4 ⊢ (𝑡 = 𝑇 → (𝑣 ∈ (mVR‘𝑡) ↦ 〈((mType‘𝑡)‘𝑣), 〈“𝑣”〉〉) = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉)) |
11 | df-mvh 32734 | . . . 4 ⊢ mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ 〈((mType‘𝑡)‘𝑣), 〈“𝑣”〉〉)) | |
12 | 10, 11, 3 | mptfvmpt 6984 | . . 3 ⊢ (𝑇 ∈ V → (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉)) |
13 | mpt0 6484 | . . . . 5 ⊢ (𝑣 ∈ ∅ ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) = ∅ | |
14 | 13 | eqcomi 2830 | . . . 4 ⊢ ∅ = (𝑣 ∈ ∅ ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
15 | fvprc 6657 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mVH‘𝑇) = ∅) | |
16 | fvprc 6657 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mVR‘𝑇) = ∅) | |
17 | 3, 16 | syl5eq 2868 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑉 = ∅) |
18 | 17 | mpteq1d 5147 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) = (𝑣 ∈ ∅ ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉)) |
19 | 14, 15, 18 | 3eqtr4a 2882 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉)) |
20 | 12, 19 | pm2.61i 184 | . 2 ⊢ (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
21 | 1, 20 | eqtri 2844 | 1 ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 〈cop 4566 ↦ cmpt 5138 ‘cfv 6349 〈“cs1 13943 mVRcmvar 32703 mTypecmty 32704 mVHcmvh 32714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-mvh 32734 |
This theorem is referenced by: mvhval 32776 mvhf 32800 |
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