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Mirrors > Home > MPE Home > Th. List > Mathboxes > mexval | Structured version Visualization version GIF version |
Description: The set of expressions, which are pairs whose first element is a typecode, and whose second element is a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mexval.k | ⊢ 𝐾 = (mTC‘𝑇) |
mexval.e | ⊢ 𝐸 = (mEx‘𝑇) |
mexval.r | ⊢ 𝑅 = (mREx‘𝑇) |
Ref | Expression |
---|---|
mexval | ⊢ 𝐸 = (𝐾 × 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mexval.e | . 2 ⊢ 𝐸 = (mEx‘𝑇) | |
2 | fveq2 6103 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇)) | |
3 | mexval.k | . . . . . 6 ⊢ 𝐾 = (mTC‘𝑇) | |
4 | 2, 3 | syl6eqr 2662 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾) |
5 | fveq2 6103 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇)) | |
6 | mexval.r | . . . . . 6 ⊢ 𝑅 = (mREx‘𝑇) | |
7 | 5, 6 | syl6eqr 2662 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅) |
8 | 4, 7 | xpeq12d 5064 | . . . 4 ⊢ (𝑡 = 𝑇 → ((mTC‘𝑡) × (mREx‘𝑡)) = (𝐾 × 𝑅)) |
9 | df-mex 30638 | . . . 4 ⊢ mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡))) | |
10 | fvex 6113 | . . . . 5 ⊢ (mTC‘𝑡) ∈ V | |
11 | fvex 6113 | . . . . 5 ⊢ (mREx‘𝑡) ∈ V | |
12 | 10, 11 | xpex 6860 | . . . 4 ⊢ ((mTC‘𝑡) × (mREx‘𝑡)) ∈ V |
13 | 8, 9, 12 | fvmpt3i 6196 | . . 3 ⊢ (𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅)) |
14 | xp0 5471 | . . . . 5 ⊢ (𝐾 × ∅) = ∅ | |
15 | 14 | eqcomi 2619 | . . . 4 ⊢ ∅ = (𝐾 × ∅) |
16 | fvprc 6097 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = ∅) | |
17 | fvprc 6097 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mREx‘𝑇) = ∅) | |
18 | 6, 17 | syl5eq 2656 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅) |
19 | 18 | xpeq2d 5063 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (𝐾 × 𝑅) = (𝐾 × ∅)) |
20 | 15, 16, 19 | 3eqtr4a 2670 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅)) |
21 | 13, 20 | pm2.61i 175 | . 2 ⊢ (mEx‘𝑇) = (𝐾 × 𝑅) |
22 | 1, 21 | eqtri 2632 | 1 ⊢ 𝐸 = (𝐾 × 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 × cxp 5036 ‘cfv 5804 mTCcmtc 30615 mRExcmrex 30617 mExcmex 30618 |
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-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
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-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 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-iota 5768 df-fun 5806 df-fv 5812 df-mex 30638 |
This theorem is referenced by: mexval2 30654 msubff 30681 msubco 30682 msubff1 30707 mvhf 30709 msubvrs 30711 |
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