MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  idaf Structured version   Visualization version   GIF version

Theorem idaf 18130
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 5485 . . 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 2740 . . 3 (Id‘𝐶) = (Id‘𝐶)
73, 4, 5, 6idafval 18124 . 2 (𝜑𝐼 = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩))
8 idaf.a . . . 4 𝐴 = (Arrow‘𝐶)
9 eqid 2740 . . . 4 (Homa𝐶) = (Homa𝐶)
108, 9homarw 18113 . . 3 (𝑥(Homa𝐶)𝑥) ⊆ 𝐴
115adantr 480 . . . 4 ((𝜑𝑥𝐵) → 𝐶 ∈ Cat)
12 simpr 484 . . . 4 ((𝜑𝑥𝐵) → 𝑥𝐵)
133, 4, 11, 12, 9idahom 18127 . . 3 ((𝜑𝑥𝐵) → (𝐼𝑥) ∈ (𝑥(Homa𝐶)𝑥))
1410, 13sselid 4006 . 2 ((𝜑𝑥𝐵) → (𝐼𝑥) ∈ 𝐴)
152, 7, 14fmpt2d 7158 1 (𝜑𝐼:𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2108  Vcvv 3488  cotp 4656  wf 6569  cfv 6573  (class class class)co 7448  Basecbs 17258  Catccat 17722  Idccid 17723  Arrowcarw 18089  Homachoma 18090  Idacida 18120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-ot 4657  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-cat 17726  df-cid 17727  df-homa 18093  df-arw 18094  df-ida 18122
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator