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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 6103 | . . 3 ⊢ (𝑣 = 𝑋 → (𝑌‘𝑣) = (𝑌‘𝑋)) | |
2 | s1eq 13233 | . . 3 ⊢ (𝑣 = 𝑋 → 〈“𝑣”〉 = 〈“𝑋”〉) | |
3 | 1, 2 | opeq12d 4348 | . 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 30684 | . 2 ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
8 | opex 4859 | . 2 ⊢ 〈(𝑌‘𝑋), 〈“𝑋”〉〉 ∈ V | |
9 | 3, 7, 8 | fvmpt 6191 | 1 ⊢ (𝑋 ∈ 𝑉 → (𝐻‘𝑋) = 〈(𝑌‘𝑋), 〈“𝑋”〉〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 〈cop 4131 ‘cfv 5804 〈“cs1 13149 mVRcmvar 30612 mTypecmty 30613 mVHcmvh 30623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-s1 13157 df-mvh 30643 |
This theorem is referenced by: mvhf1 30710 msubvrs 30711 |
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