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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 6897 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇)) | |
2 | 1 | rneqd 5940 | . . . 4 ⊢ (𝑡 = 𝑇 → ran (mType‘𝑡) = ran (mType‘𝑇)) |
3 | df-mvt 35095 | . . . 4 ⊢ mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡)) | |
4 | fvex 6910 | . . . . 5 ⊢ (mType‘𝑇) ∈ V | |
5 | 4 | rnex 7918 | . . . 4 ⊢ ran (mType‘𝑇) ∈ V |
6 | 2, 3, 5 | fvmpt 7005 | . . 3 ⊢ (𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇)) |
7 | rn0 5928 | . . . . 5 ⊢ ran ∅ = ∅ | |
8 | 7 | eqcomi 2737 | . . . 4 ⊢ ∅ = ran ∅ |
9 | fvprc 6889 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mVT‘𝑇) = ∅) | |
10 | fvprc 6889 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → (mType‘𝑇) = ∅) | |
11 | 10 | rneqd 5940 | . . . 4 ⊢ (¬ 𝑇 ∈ V → ran (mType‘𝑇) = ran ∅) |
12 | 8, 9, 11 | 3eqtr4a 2794 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇)) |
13 | 6, 12 | pm2.61i 182 | . 2 ⊢ (mVT‘𝑇) = ran (mType‘𝑇) |
14 | mvtval.f | . 2 ⊢ 𝑉 = (mVT‘𝑇) | |
15 | mvtval.y | . . 3 ⊢ 𝑌 = (mType‘𝑇) | |
16 | 15 | rneqi 5939 | . 2 ⊢ ran 𝑌 = ran (mType‘𝑇) |
17 | 13, 14, 16 | 3eqtr4i 2766 | 1 ⊢ 𝑉 = ran 𝑌 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ∅c0 4323 ran crn 5679 ‘cfv 6548 mTypecmty 35072 mVTcmvt 35073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6500 df-fun 6550 df-fv 6556 df-mvt 35095 |
This theorem is referenced by: mtyf 35162 mvtss 35163 |
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