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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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