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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 6645 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇)) | |
2 | 1 | rneqd 5772 | . . . 4 ⊢ (𝑡 = 𝑇 → ran (mType‘𝑡) = ran (mType‘𝑇)) |
3 | df-mvt 32845 | . . . 4 ⊢ mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡)) | |
4 | fvex 6658 | . . . . 5 ⊢ (mType‘𝑇) ∈ V | |
5 | 4 | rnex 7599 | . . . 4 ⊢ ran (mType‘𝑇) ∈ V |
6 | 2, 3, 5 | fvmpt 6745 | . . 3 ⊢ (𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇)) |
7 | rn0 5760 | . . . . 5 ⊢ ran ∅ = ∅ | |
8 | 7 | eqcomi 2807 | . . . 4 ⊢ ∅ = ran ∅ |
9 | fvprc 6638 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mVT‘𝑇) = ∅) | |
10 | fvprc 6638 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → (mType‘𝑇) = ∅) | |
11 | 10 | rneqd 5772 | . . . 4 ⊢ (¬ 𝑇 ∈ V → ran (mType‘𝑇) = ran ∅) |
12 | 8, 9, 11 | 3eqtr4a 2859 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mVT‘𝑇) = ran (mType‘𝑇)) |
13 | 6, 12 | pm2.61i 185 | . 2 ⊢ (mVT‘𝑇) = ran (mType‘𝑇) |
14 | mvtval.f | . 2 ⊢ 𝑉 = (mVT‘𝑇) | |
15 | mvtval.y | . . 3 ⊢ 𝑌 = (mType‘𝑇) | |
16 | 15 | rneqi 5771 | . 2 ⊢ ran 𝑌 = ran (mType‘𝑇) |
17 | 13, 14, 16 | 3eqtr4i 2831 | 1 ⊢ 𝑉 = ran 𝑌 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 ran crn 5520 ‘cfv 6324 mTypecmty 32822 mVTcmvt 32823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-mvt 32845 |
This theorem is referenced by: mtyf 32912 mvtss 32913 |
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