Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tendo02 Structured version   Visualization version   GIF version

Theorem tendo02 40744
Description: Value of additive identity endomorphism. (Contributed by NM, 11-Jun-2013.)
Hypotheses
Ref Expression
tendo0cbv.o 𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
tendo02.b 𝐵 = (Base‘𝐾)
Assertion
Ref Expression
tendo02 (𝐹𝑇 → (𝑂𝐹) = ( I ↾ 𝐵))
Distinct variable groups:   𝐵,𝑓   𝑇,𝑓
Allowed substitution hints:   𝐹(𝑓)   𝐾(𝑓)   𝑂(𝑓)

Proof of Theorem tendo02
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 eqidd 2741 . 2 (𝑔 = 𝐹 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
2 tendo0cbv.o . . 3 𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
32tendo0cbv 40743 . 2 𝑂 = (𝑔𝑇 ↦ ( I ↾ 𝐵))
4 funi 6610 . . 3 Fun I
5 tendo02.b . . . 4 𝐵 = (Base‘𝐾)
65fvexi 6934 . . 3 𝐵 ∈ V
7 resfunexg 7252 . . 3 ((Fun I ∧ 𝐵 ∈ V) → ( I ↾ 𝐵) ∈ V)
84, 6, 7mp2an 691 . 2 ( I ↾ 𝐵) ∈ V
91, 3, 8fvmpt 7029 1 (𝐹𝑇 → (𝑂𝐹) = ( I ↾ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2108  Vcvv 3488  cmpt 5249   I cid 5592  cres 5702  Fun wfun 6567  cfv 6573  Basecbs 17258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581
This theorem is referenced by:  tendo0co2  40745  tendo0tp  40746  tendo0pl  40748  tendoipl  40754  tendoid0  40782  tendo0mul  40783  tendo0mulr  40784  tendo1ne0  40785  tendoex  40932  dicn0  41149  dihordlem7b  41172  dihmeetlem1N  41247  dihglblem5apreN  41248  dihmeetlem4preN  41263  dihmeetlem13N  41276
  Copyright terms: Public domain W3C validator