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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 6191 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇)) | |
3 | mexval.k | . . . . . 6 ⊢ 𝐾 = (mTC‘𝑇) | |
4 | 2, 3 | syl6eqr 2674 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾) |
5 | fveq2 6191 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇)) | |
6 | mexval.r | . . . . . 6 ⊢ 𝑅 = (mREx‘𝑇) | |
7 | 5, 6 | syl6eqr 2674 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅) |
8 | 4, 7 | xpeq12d 5140 | . . . 4 ⊢ (𝑡 = 𝑇 → ((mTC‘𝑡) × (mREx‘𝑡)) = (𝐾 × 𝑅)) |
9 | df-mex 31384 | . . . 4 ⊢ mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡))) | |
10 | fvex 6201 | . . . . 5 ⊢ (mTC‘𝑡) ∈ V | |
11 | fvex 6201 | . . . . 5 ⊢ (mREx‘𝑡) ∈ V | |
12 | 10, 11 | xpex 6962 | . . . 4 ⊢ ((mTC‘𝑡) × (mREx‘𝑡)) ∈ V |
13 | 8, 9, 12 | fvmpt3i 6287 | . . 3 ⊢ (𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅)) |
14 | xp0 5552 | . . . . 5 ⊢ (𝐾 × ∅) = ∅ | |
15 | 14 | eqcomi 2631 | . . . 4 ⊢ ∅ = (𝐾 × ∅) |
16 | fvprc 6185 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = ∅) | |
17 | fvprc 6185 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mREx‘𝑇) = ∅) | |
18 | 6, 17 | syl5eq 2668 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅) |
19 | 18 | xpeq2d 5139 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (𝐾 × 𝑅) = (𝐾 × ∅)) |
20 | 15, 16, 19 | 3eqtr4a 2682 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅)) |
21 | 13, 20 | pm2.61i 176 | . 2 ⊢ (mEx‘𝑇) = (𝐾 × 𝑅) |
22 | 1, 21 | eqtri 2644 | 1 ⊢ 𝐸 = (𝐾 × 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 × cxp 5112 ‘cfv 5888 mTCcmtc 31361 mRExcmrex 31363 mExcmex 31364 |
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-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
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-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 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-iota 5851 df-fun 5890 df-fv 5896 df-mex 31384 |
This theorem is referenced by: mexval2 31400 msubff 31427 msubco 31428 msubff1 31453 mvhf 31455 msubvrs 31457 |
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