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Theorem tendo02 40292
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 2729 . 2 (𝑔 = 𝐹 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
2 tendo0cbv.o . . 3 𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
32tendo0cbv 40291 . 2 𝑂 = (𝑔𝑇 ↦ ( I ↾ 𝐵))
4 funi 6590 . . 3 Fun I
5 tendo02.b . . . 4 𝐵 = (Base‘𝐾)
65fvexi 6916 . . 3 𝐵 ∈ V
7 resfunexg 7233 . . 3 ((Fun I ∧ 𝐵 ∈ V) → ( I ↾ 𝐵) ∈ V)
84, 6, 7mp2an 690 . 2 ( I ↾ 𝐵) ∈ V
91, 3, 8fvmpt 7010 1 (𝐹𝑇 → (𝑂𝐹) = ( I ↾ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3473  cmpt 5235   I cid 5579  cres 5684  Fun wfun 6547  cfv 6553  Basecbs 17187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561
This theorem is referenced by:  tendo0co2  40293  tendo0tp  40294  tendo0pl  40296  tendoipl  40302  tendoid0  40330  tendo0mul  40331  tendo0mulr  40332  tendo1ne0  40333  tendoex  40480  dicn0  40697  dihordlem7b  40720  dihmeetlem1N  40795  dihglblem5apreN  40796  dihmeetlem4preN  40811  dihmeetlem13N  40824
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