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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 6906 | . . 3 ⊢ (𝑣 = 𝑋 → (𝑌‘𝑣) = (𝑌‘𝑋)) | |
| 2 | s1eq 14638 | . . 3 ⊢ (𝑣 = 𝑋 → 〈“𝑣”〉 = 〈“𝑋”〉) | |
| 3 | 1, 2 | opeq12d 4881 | . 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 35538 | . 2 ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
| 8 | opex 5469 | . 2 ⊢ 〈(𝑌‘𝑋), 〈“𝑋”〉〉 ∈ V | |
| 9 | 3, 7, 8 | fvmpt 7016 | 1 ⊢ (𝑋 ∈ 𝑉 → (𝐻‘𝑋) = 〈(𝑌‘𝑋), 〈“𝑋”〉〉) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 〈cop 4632 ‘cfv 6561 〈“cs1 14633 mVRcmvar 35466 mTypecmty 35467 mVHcmvh 35477 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-s1 14634 df-mvh 35497 |
| This theorem is referenced by: mvhf1 35564 msubvrs 35565 |
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