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Theorem tendo02 40170
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 2727 . 2 (𝑔 = 𝐹 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
2 tendo0cbv.o . . 3 𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
32tendo0cbv 40169 . 2 𝑂 = (𝑔𝑇 ↦ ( I ↾ 𝐵))
4 funi 6573 . . 3 Fun I
5 tendo02.b . . . 4 𝐵 = (Base‘𝐾)
65fvexi 6898 . . 3 𝐵 ∈ V
7 resfunexg 7211 . . 3 ((Fun I ∧ 𝐵 ∈ V) → ( I ↾ 𝐵) ∈ V)
84, 6, 7mp2an 689 . 2 ( I ↾ 𝐵) ∈ V
91, 3, 8fvmpt 6991 1 (𝐹𝑇 → (𝑂𝐹) = ( I ↾ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3468  cmpt 5224   I cid 5566  cres 5671  Fun wfun 6530  cfv 6536  Basecbs 17150
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544
This theorem is referenced by:  tendo0co2  40171  tendo0tp  40172  tendo0pl  40174  tendoipl  40180  tendoid0  40208  tendo0mul  40209  tendo0mulr  40210  tendo1ne0  40211  tendoex  40358  dicn0  40575  dihordlem7b  40598  dihmeetlem1N  40673  dihglblem5apreN  40674  dihmeetlem4preN  40689  dihmeetlem13N  40702
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