Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > mvtval | Structured version Visualization version GIF version |
Description: The set of variable typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mvtval.f | ⊢ 𝑉 = (mVT‘𝑇) |
mvtval.y | ⊢ 𝑌 = (mType‘𝑇) |
Ref | Expression |
---|---|
mvtval | ⊢ 𝑉 = ran 𝑌 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6103 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇)) | |
2 | 1 | rneqd 5274 | . . . 4 ⊢ (𝑡 = 𝑇 → ran (mType‘𝑡) = ran (mType‘𝑇)) |
3 | df-mvt 30636 | . . . 4 ⊢ mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡)) | |
4 | fvex 6113 | . . . . 5 ⊢ (mType‘𝑇) ∈ V | |
5 | 4 | rnex 6992 | . . . 4 ⊢ ran (mType‘𝑇) ∈ V |
6 | 2, 3, 5 | fvmpt 6191 | . . 3 ⊢ (𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇)) |
7 | rn0 5298 | . . . . 5 ⊢ ran ∅ = ∅ | |
8 | 7 | eqcomi 2619 | . . . 4 ⊢ ∅ = ran ∅ |
9 | fvprc 6097 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mVT‘𝑇) = ∅) | |
10 | fvprc 6097 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → (mType‘𝑇) = ∅) | |
11 | 10 | rneqd 5274 | . . . 4 ⊢ (¬ 𝑇 ∈ V → ran (mType‘𝑇) = ran ∅) |
12 | 8, 9, 11 | 3eqtr4a 2670 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇)) |
13 | 6, 12 | pm2.61i 175 | . 2 ⊢ (mVT‘𝑇) = ran (mType‘𝑇) |
14 | mvtval.f | . 2 ⊢ 𝑉 = (mVT‘𝑇) | |
15 | mvtval.y | . . 3 ⊢ 𝑌 = (mType‘𝑇) | |
16 | 15 | rneqi 5273 | . 2 ⊢ ran 𝑌 = ran (mType‘𝑇) |
17 | 13, 14, 16 | 3eqtr4i 2642 | 1 ⊢ 𝑉 = ran 𝑌 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ran crn 5039 ‘cfv 5804 mTypecmty 30613 mVTcmvt 30614 |
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-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-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-mvt 30636 |
This theorem is referenced by: mtyf 30703 mvtss 30704 |
Copyright terms: Public domain | W3C validator |