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Theorem idaf 16914
 Description: The identity arrow function is a function from objects to arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
idafval.i 𝐼 = (Ida𝐶)
idafval.b 𝐵 = (Base‘𝐶)
idafval.c (𝜑𝐶 ∈ Cat)
idaf.a 𝐴 = (Arrow‘𝐶)
Assertion
Ref Expression
idaf (𝜑𝐼:𝐵𝐴)

Proof of Theorem idaf
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 otex 5082 . . 3 𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ V
21a1i 11 . 2 ((𝜑𝑥𝐵) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ V)
3 idafval.i . . 3 𝐼 = (Ida𝐶)
4 idafval.b . . 3 𝐵 = (Base‘𝐶)
5 idafval.c . . 3 (𝜑𝐶 ∈ Cat)
6 eqid 2760 . . 3 (Id‘𝐶) = (Id‘𝐶)
73, 4, 5, 6idafval 16908 . 2 (𝜑𝐼 = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩))
8 idaf.a . . . 4 𝐴 = (Arrow‘𝐶)
9 eqid 2760 . . . 4 (Homa𝐶) = (Homa𝐶)
108, 9homarw 16897 . . 3 (𝑥(Homa𝐶)𝑥) ⊆ 𝐴
115adantr 472 . . . 4 ((𝜑𝑥𝐵) → 𝐶 ∈ Cat)
12 simpr 479 . . . 4 ((𝜑𝑥𝐵) → 𝑥𝐵)
133, 4, 11, 12, 9idahom 16911 . . 3 ((𝜑𝑥𝐵) → (𝐼𝑥) ∈ (𝑥(Homa𝐶)𝑥))
1410, 13sseldi 3742 . 2 ((𝜑𝑥𝐵) → (𝐼𝑥) ∈ 𝐴)
152, 7, 14fmpt2d 6556 1 (𝜑𝐼:𝐵𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  Vcvv 3340  ⟨cotp 4329  ⟶wf 6045  ‘cfv 6049  (class class class)co 6813  Basecbs 16059  Catccat 16526  Idccid 16527  Arrowcarw 16873  Homachoma 16874  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-rmo 3058  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-pw 4304  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-riota 6774  df-ov 6816  df-cat 16530  df-cid 16531  df-homa 16877  df-arw 16878  df-ida 16906 This theorem is referenced by: (None)
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