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Theorem ida2 17385
 Description: Morphism part of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
idafval.i 𝐼 = (Ida𝐶)
idafval.b 𝐵 = (Base‘𝐶)
idafval.c (𝜑𝐶 ∈ Cat)
idafval.1 1 = (Id‘𝐶)
idaval.x (𝜑𝑋𝐵)
Assertion
Ref Expression
ida2 (𝜑 → (2nd ‘(𝐼𝑋)) = ( 1𝑋))

Proof of Theorem ida2
StepHypRef Expression
1 idafval.i . . . 4 𝐼 = (Ida𝐶)
2 idafval.b . . . 4 𝐵 = (Base‘𝐶)
3 idafval.c . . . 4 (𝜑𝐶 ∈ Cat)
4 idafval.1 . . . 4 1 = (Id‘𝐶)
5 idaval.x . . . 4 (𝜑𝑋𝐵)
61, 2, 3, 4, 5idaval 17384 . . 3 (𝜑 → (𝐼𝑋) = ⟨𝑋, 𝑋, ( 1𝑋)⟩)
76fveq2d 6662 . 2 (𝜑 → (2nd ‘(𝐼𝑋)) = (2nd ‘⟨𝑋, 𝑋, ( 1𝑋)⟩))
8 fvex 6671 . . 3 ( 1𝑋) ∈ V
9 ot3rdg 7709 . . 3 (( 1𝑋) ∈ V → (2nd ‘⟨𝑋, 𝑋, ( 1𝑋)⟩) = ( 1𝑋))
108, 9ax-mp 5 . 2 (2nd ‘⟨𝑋, 𝑋, ( 1𝑋)⟩) = ( 1𝑋)
117, 10eqtrdi 2809 1 (𝜑 → (2nd ‘(𝐼𝑋)) = ( 1𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ⟨cotp 4530  ‘cfv 6335  2nd c2nd 7692  Basecbs 16541  Catccat 16993  Idccid 16994  Idacida 17379 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-ot 4531  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-2nd 7694  df-ida 17381 This theorem is referenced by:  arwlid  17398  arwrid  17399
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