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 6668 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇)) | |
3 | mvhfval.v | . . . . . 6 ⊢ 𝑉 = (mVR‘𝑇) | |
4 | 2, 3 | eqtr4di 2791 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉) |
5 | fveq2 6668 | . . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇)) | |
6 | mvhfval.y | . . . . . . . 8 ⊢ 𝑌 = (mType‘𝑇) | |
7 | 5, 6 | eqtr4di 2791 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = 𝑌) |
8 | 7 | fveq1d 6670 | . . . . . 6 ⊢ (𝑡 = 𝑇 → ((mType‘𝑡)‘𝑣) = (𝑌‘𝑣)) |
9 | 8 | opeq1d 4764 | . . . . 5 ⊢ (𝑡 = 𝑇 → 〈((mType‘𝑡)‘𝑣), 〈“𝑣”〉〉 = 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
10 | 4, 9 | mpteq12dv 5112 | . . . 4 ⊢ (𝑡 = 𝑇 → (𝑣 ∈ (mVR‘𝑡) ↦ 〈((mType‘𝑡)‘𝑣), 〈“𝑣”〉〉) = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉)) |
11 | df-mvh 33017 | . . . 4 ⊢ mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ 〈((mType‘𝑡)‘𝑣), 〈“𝑣”〉〉)) | |
12 | 10, 11, 3 | mptfvmpt 6995 | . . 3 ⊢ (𝑇 ∈ V → (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉)) |
13 | mpt0 6473 | . . . . 5 ⊢ (𝑣 ∈ ∅ ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) = ∅ | |
14 | 13 | eqcomi 2747 | . . . 4 ⊢ ∅ = (𝑣 ∈ ∅ ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
15 | fvprc 6660 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mVH‘𝑇) = ∅) | |
16 | fvprc 6660 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mVR‘𝑇) = ∅) | |
17 | 3, 16 | syl5eq 2785 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑉 = ∅) |
18 | 17 | mpteq1d 5116 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) = (𝑣 ∈ ∅ ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉)) |
19 | 14, 15, 18 | 3eqtr4a 2799 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉)) |
20 | 12, 19 | pm2.61i 185 | . 2 ⊢ (mVH‘𝑇) = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
21 | 1, 20 | eqtri 2761 | 1 ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2113 Vcvv 3397 ∅c0 4209 〈cop 4519 ↦ cmpt 5107 ‘cfv 6333 〈“cs1 14031 mVRcmvar 32986 mTypecmty 32987 mVHcmvh 32997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-mvh 33017 |
This theorem is referenced by: mvhval 33059 mvhf 33083 |
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