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Theorem idaval 17307
Description: Value of the identity arrow function. (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
idaval (𝜑 → (𝐼𝑋) = ⟨𝑋, 𝑋, ( 1𝑋)⟩)

Proof of Theorem idaval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 idafval.i . . 3 𝐼 = (Ida𝐶)
2 idafval.b . . 3 𝐵 = (Base‘𝐶)
3 idafval.c . . 3 (𝜑𝐶 ∈ Cat)
4 idafval.1 . . 3 1 = (Id‘𝐶)
51, 2, 3, 4idafval 17306 . 2 (𝜑𝐼 = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩))
6 simpr 488 . . 3 ((𝜑𝑥 = 𝑋) → 𝑥 = 𝑋)
76fveq2d 6655 . . 3 ((𝜑𝑥 = 𝑋) → ( 1𝑥) = ( 1𝑋))
86, 6, 7oteq123d 4799 . 2 ((𝜑𝑥 = 𝑋) → ⟨𝑥, 𝑥, ( 1𝑥)⟩ = ⟨𝑋, 𝑋, ( 1𝑋)⟩)
9 idaval.x . 2 (𝜑𝑋𝐵)
10 otex 5338 . . 3 𝑋, 𝑋, ( 1𝑋)⟩ ∈ V
1110a1i 11 . 2 (𝜑 → ⟨𝑋, 𝑋, ( 1𝑋)⟩ ∈ V)
125, 8, 9, 11fvmptd 6756 1 (𝜑 → (𝐼𝑋) = ⟨𝑋, 𝑋, ( 1𝑋)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3479  cotp 4556  cfv 6336  Basecbs 16472  Catccat 16924  Idccid 16925  Idacida 17302
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-ot 4557  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ida 17304
This theorem is referenced by:  ida2  17308  idahom  17309
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