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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 6191 | . . 3 ⊢ (𝑣 = 𝑋 → (𝑌‘𝑣) = (𝑌‘𝑋)) | |
2 | s1eq 13380 | . . 3 ⊢ (𝑣 = 𝑋 → 〈“𝑣”〉 = 〈“𝑋”〉) | |
3 | 1, 2 | opeq12d 4410 | . 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 31430 | . 2 ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) |
8 | opex 4932 | . 2 ⊢ 〈(𝑌‘𝑋), 〈“𝑋”〉〉 ∈ V | |
9 | 3, 7, 8 | fvmpt 6282 | 1 ⊢ (𝑋 ∈ 𝑉 → (𝐻‘𝑋) = 〈(𝑌‘𝑋), 〈“𝑋”〉〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 〈cop 4183 ‘cfv 5888 〈“cs1 13294 mVRcmvar 31358 mTypecmty 31359 mVHcmvh 31369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-s1 13302 df-mvh 31389 |
This theorem is referenced by: mvhf1 31456 msubvrs 31457 |
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