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Mathbox for Mario Carneiro |
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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 6645 | . . 3 ⊢ (𝑣 = 𝑋 → (𝑌‘𝑣) = (𝑌‘𝑋)) | |
2 | s1eq 13945 | . . 3 ⊢ (𝑣 = 𝑋 → 〈“𝑣”〉 = 〈“𝑋”〉) | |
3 | 1, 2 | opeq12d 4773 | . 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 32893 | . 2 ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
8 | opex 5321 | . 2 ⊢ 〈(𝑌‘𝑋), 〈“𝑋”〉〉 ∈ V | |
9 | 3, 7, 8 | fvmpt 6745 | 1 ⊢ (𝑋 ∈ 𝑉 → (𝐻‘𝑋) = 〈(𝑌‘𝑋), 〈“𝑋”〉〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 〈cop 4531 ‘cfv 6324 〈“cs1 13940 mVRcmvar 32821 mTypecmty 32822 mVHcmvh 32832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 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 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-s1 13941 df-mvh 32852 |
This theorem is referenced by: mvhf1 32919 msubvrs 32920 |
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