Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > mvtinf | Structured version Visualization version GIF version |
Description: Each variable typecode has infinitely many variables. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mvtinf.f | ⊢ 𝐹 = (mVT‘𝑇) |
mvtinf.y | ⊢ 𝑌 = (mType‘𝑇) |
Ref | Expression |
---|---|
mvtinf | ⊢ ((𝑇 ∈ mFS ∧ 𝑋 ∈ 𝐹) → ¬ (◡𝑌 “ {𝑋}) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . . 5 ⊢ (mCN‘𝑇) = (mCN‘𝑇) | |
2 | eqid 2821 | . . . . 5 ⊢ (mVR‘𝑇) = (mVR‘𝑇) | |
3 | mvtinf.y | . . . . 5 ⊢ 𝑌 = (mType‘𝑇) | |
4 | mvtinf.f | . . . . 5 ⊢ 𝐹 = (mVT‘𝑇) | |
5 | eqid 2821 | . . . . 5 ⊢ (mTC‘𝑇) = (mTC‘𝑇) | |
6 | eqid 2821 | . . . . 5 ⊢ (mAx‘𝑇) = (mAx‘𝑇) | |
7 | eqid 2821 | . . . . 5 ⊢ (mStat‘𝑇) = (mStat‘𝑇) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ismfs 32796 | . . . 4 ⊢ (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ 𝑌:(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin)))) |
9 | 8 | ibi 269 | . . 3 ⊢ (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ 𝑌:(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin))) |
10 | 9 | simprrd 772 | . 2 ⊢ (𝑇 ∈ mFS → ∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin) |
11 | sneq 4577 | . . . . . 6 ⊢ (𝑣 = 𝑋 → {𝑣} = {𝑋}) | |
12 | 11 | imaeq2d 5929 | . . . . 5 ⊢ (𝑣 = 𝑋 → (◡𝑌 “ {𝑣}) = (◡𝑌 “ {𝑋})) |
13 | 12 | eleq1d 2897 | . . . 4 ⊢ (𝑣 = 𝑋 → ((◡𝑌 “ {𝑣}) ∈ Fin ↔ (◡𝑌 “ {𝑋}) ∈ Fin)) |
14 | 13 | notbid 320 | . . 3 ⊢ (𝑣 = 𝑋 → (¬ (◡𝑌 “ {𝑣}) ∈ Fin ↔ ¬ (◡𝑌 “ {𝑋}) ∈ Fin)) |
15 | 14 | rspccva 3622 | . 2 ⊢ ((∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin ∧ 𝑋 ∈ 𝐹) → ¬ (◡𝑌 “ {𝑋}) ∈ Fin) |
16 | 10, 15 | sylan 582 | 1 ⊢ ((𝑇 ∈ mFS ∧ 𝑋 ∈ 𝐹) → ¬ (◡𝑌 “ {𝑋}) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 {csn 4567 ◡ccnv 5554 “ cima 5558 ⟶wf 6351 ‘cfv 6355 Fincfn 8509 mCNcmcn 32707 mVRcmvar 32708 mTypecmty 32709 mVTcmvt 32710 mTCcmtc 32711 mAxcmax 32712 mStatcmsta 32722 mFScmfs 32723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-mfs 32743 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |