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Mirrors > Home > MPE Home > Th. List > Mathboxes > mvhval | 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 |
---|---|
mvhval | ⊢ (𝑋 ∈ 𝑉 → (𝐻‘𝑋) = 〈(𝑌‘𝑋), 〈“𝑋”〉〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6663 | . . 3 ⊢ (𝑣 = 𝑋 → (𝑌‘𝑣) = (𝑌‘𝑋)) | |
2 | s1eq 13942 | . . 3 ⊢ (𝑣 = 𝑋 → 〈“𝑣”〉 = 〈“𝑋”〉) | |
3 | 1, 2 | opeq12d 4803 | . 2 ⊢ (𝑣 = 𝑋 → 〈(𝑌‘𝑣), 〈“𝑣”〉〉 = 〈(𝑌‘𝑋), 〈“𝑋”〉〉) |
4 | mvhfval.v | . . 3 ⊢ 𝑉 = (mVR‘𝑇) | |
5 | mvhfval.y | . . 3 ⊢ 𝑌 = (mType‘𝑇) | |
6 | mvhfval.h | . . 3 ⊢ 𝐻 = (mVH‘𝑇) | |
7 | 4, 5, 6 | mvhfval 32677 | . 2 ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
8 | opex 5347 | . 2 ⊢ 〈(𝑌‘𝑋), 〈“𝑋”〉〉 ∈ V | |
9 | 3, 7, 8 | fvmpt 6761 | 1 ⊢ (𝑋 ∈ 𝑉 → (𝐻‘𝑋) = 〈(𝑌‘𝑋), 〈“𝑋”〉〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 〈cop 4563 ‘cfv 6348 〈“cs1 13937 mVRcmvar 32605 mTypecmty 32606 mVHcmvh 32616 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-s1 13938 df-mvh 32636 |
This theorem is referenced by: mvhf1 32703 msubvrs 32704 |
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