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Mirrors > Home > MPE Home > Th. List > Mathboxes > mtyf | Structured version Visualization version GIF version |
Description: The type function maps variables to variable typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mtyf.v | ⊢ 𝑉 = (mVR‘𝑇) |
mtyf.f | ⊢ 𝐹 = (mVT‘𝑇) |
mtyf.y | ⊢ 𝑌 = (mType‘𝑇) |
Ref | Expression |
---|---|
mtyf | ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mtyf.v | . . . 4 ⊢ 𝑉 = (mVR‘𝑇) | |
2 | eqid 2610 | . . . 4 ⊢ (mTC‘𝑇) = (mTC‘𝑇) | |
3 | mtyf.y | . . . 4 ⊢ 𝑌 = (mType‘𝑇) | |
4 | 1, 2, 3 | mtyf2 30702 | . . 3 ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶(mTC‘𝑇)) |
5 | ffn 5958 | . . . 4 ⊢ (𝑌:𝑉⟶(mTC‘𝑇) → 𝑌 Fn 𝑉) | |
6 | dffn4 6034 | . . . 4 ⊢ (𝑌 Fn 𝑉 ↔ 𝑌:𝑉–onto→ran 𝑌) | |
7 | 5, 6 | sylib 207 | . . 3 ⊢ (𝑌:𝑉⟶(mTC‘𝑇) → 𝑌:𝑉–onto→ran 𝑌) |
8 | fof 6028 | . . 3 ⊢ (𝑌:𝑉–onto→ran 𝑌 → 𝑌:𝑉⟶ran 𝑌) | |
9 | 4, 7, 8 | 3syl 18 | . 2 ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶ran 𝑌) |
10 | mtyf.f | . . . 4 ⊢ 𝐹 = (mVT‘𝑇) | |
11 | 10, 3 | mvtval 30651 | . . 3 ⊢ 𝐹 = ran 𝑌 |
12 | feq3 5941 | . . 3 ⊢ (𝐹 = ran 𝑌 → (𝑌:𝑉⟶𝐹 ↔ 𝑌:𝑉⟶ran 𝑌)) | |
13 | 11, 12 | ax-mp 5 | . 2 ⊢ (𝑌:𝑉⟶𝐹 ↔ 𝑌:𝑉⟶ran 𝑌) |
14 | 9, 13 | sylibr 223 | 1 ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ran crn 5039 Fn wfn 5799 ⟶wf 5800 –onto→wfo 5802 ‘cfv 5804 mVRcmvar 30612 mTypecmty 30613 mVTcmvt 30614 mTCcmtc 30615 mFScmfs 30627 |
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-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-mvt 30636 df-mfs 30647 |
This theorem is referenced by: (None) |
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