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Theorem ida2 18013
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 18012 . . 3 (𝜑 → (𝐼𝑋) = ⟨𝑋, 𝑋, ( 1𝑋)⟩)
76fveq2d 6886 . 2 (𝜑 → (2nd ‘(𝐼𝑋)) = (2nd ‘⟨𝑋, 𝑋, ( 1𝑋)⟩))
8 fvex 6895 . . 3 ( 1𝑋) ∈ V
9 ot3rdg 7985 . . 3 (( 1𝑋) ∈ V → (2nd ‘⟨𝑋, 𝑋, ( 1𝑋)⟩) = ( 1𝑋))
108, 9ax-mp 5 . 2 (2nd ‘⟨𝑋, 𝑋, ( 1𝑋)⟩) = ( 1𝑋)
117, 10eqtrdi 2780 1 (𝜑 → (2nd ‘(𝐼𝑋)) = ( 1𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3466  cotp 4629  cfv 6534  2nd c2nd 7968  Basecbs 17145  Catccat 17609  Idccid 17610  Idacida 18007
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 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-ot 4630  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-2nd 7970  df-ida 18009
This theorem is referenced by:  arwlid  18026  arwrid  18027
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