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Theorem ida2 17148
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 17147 . . 3 (𝜑 → (𝐼𝑋) = ⟨𝑋, 𝑋, ( 1𝑋)⟩)
76fveq2d 6542 . 2 (𝜑 → (2nd ‘(𝐼𝑋)) = (2nd ‘⟨𝑋, 𝑋, ( 1𝑋)⟩))
8 fvex 6551 . . 3 ( 1𝑋) ∈ V
9 ot3rdg 7561 . . 3 (( 1𝑋) ∈ V → (2nd ‘⟨𝑋, 𝑋, ( 1𝑋)⟩) = ( 1𝑋))
108, 9ax-mp 5 . 2 (2nd ‘⟨𝑋, 𝑋, ( 1𝑋)⟩) = ( 1𝑋)
117, 10syl6eq 2847 1 (𝜑 → (2nd ‘(𝐼𝑋)) = ( 1𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1522  wcel 2081  Vcvv 3437  cotp 4480  cfv 6225  2nd c2nd 7544  Basecbs 16312  Catccat 16764  Idccid 16765  Idacida 17142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-ot 4481  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-2nd 7546  df-ida 17144
This theorem is referenced by:  arwlid  17161  arwrid  17162
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