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Theorem ida2 16910
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 16909 . . 3 (𝜑 → (𝐼𝑋) = ⟨𝑋, 𝑋, ( 1𝑋)⟩)
76fveq2d 6356 . 2 (𝜑 → (2nd ‘(𝐼𝑋)) = (2nd ‘⟨𝑋, 𝑋, ( 1𝑋)⟩))
8 fvex 6362 . . 3 ( 1𝑋) ∈ V
9 ot3rdg 7349 . . 3 (( 1𝑋) ∈ V → (2nd ‘⟨𝑋, 𝑋, ( 1𝑋)⟩) = ( 1𝑋))
108, 9ax-mp 5 . 2 (2nd ‘⟨𝑋, 𝑋, ( 1𝑋)⟩) = ( 1𝑋)
117, 10syl6eq 2810 1 (𝜑 → (2nd ‘(𝐼𝑋)) = ( 1𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  cotp 4329  cfv 6049  2nd c2nd 7332  Basecbs 16059  Catccat 16526  Idccid 16527  Idacida 16904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-ot 4330  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-2nd 7334  df-ida 16906
This theorem is referenced by:  arwlid  16923  arwrid  16924
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