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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 6846 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇)) | |
2 | 1 | rneqd 5897 | . . . 4 ⊢ (𝑡 = 𝑇 → ran (mType‘𝑡) = ran (mType‘𝑇)) |
3 | df-mvt 34143 | . . . 4 ⊢ mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡)) | |
4 | fvex 6859 | . . . . 5 ⊢ (mType‘𝑇) ∈ V | |
5 | 4 | rnex 7853 | . . . 4 ⊢ ran (mType‘𝑇) ∈ V |
6 | 2, 3, 5 | fvmpt 6952 | . . 3 ⊢ (𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇)) |
7 | rn0 5885 | . . . . 5 ⊢ ran ∅ = ∅ | |
8 | 7 | eqcomi 2742 | . . . 4 ⊢ ∅ = ran ∅ |
9 | fvprc 6838 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mVT‘𝑇) = ∅) | |
10 | fvprc 6838 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → (mType‘𝑇) = ∅) | |
11 | 10 | rneqd 5897 | . . . 4 ⊢ (¬ 𝑇 ∈ V → ran (mType‘𝑇) = ran ∅) |
12 | 8, 9, 11 | 3eqtr4a 2799 | . . 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 5896 | . 2 ⊢ ran 𝑌 = ran (mType‘𝑇) |
17 | 13, 14, 16 | 3eqtr4i 2771 | 1 ⊢ 𝑉 = ran 𝑌 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 Vcvv 3447 ∅c0 4286 ran crn 5638 ‘cfv 6500 mTypecmty 34120 mVTcmvt 34121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-iota 6452 df-fun 6502 df-fv 6508 df-mvt 34143 |
This theorem is referenced by: mtyf 34210 mvtss 34211 |
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