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

Theorem idaf 18117
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 5476 . . 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 2735 . . 3 (Id‘𝐶) = (Id‘𝐶)
73, 4, 5, 6idafval 18111 . 2 (𝜑𝐼 = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩))
8 idaf.a . . . 4 𝐴 = (Arrow‘𝐶)
9 eqid 2735 . . . 4 (Homa𝐶) = (Homa𝐶)
108, 9homarw 18100 . . 3 (𝑥(Homa𝐶)𝑥) ⊆ 𝐴
115adantr 480 . . . 4 ((𝜑𝑥𝐵) → 𝐶 ∈ Cat)
12 simpr 484 . . . 4 ((𝜑𝑥𝐵) → 𝑥𝐵)
133, 4, 11, 12, 9idahom 18114 . . 3 ((𝜑𝑥𝐵) → (𝐼𝑥) ∈ (𝑥(Homa𝐶)𝑥))
1410, 13sselid 3993 . 2 ((𝜑𝑥𝐵) → (𝐼𝑥) ∈ 𝐴)
152, 7, 14fmpt2d 7144 1 (𝜑𝐼:𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cotp 4639  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  Catccat 17709  Idccid 17710  Arrowcarw 18076  Homachoma 18077  Idacida 18107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-ot 4640  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-cat 17713  df-cid 17714  df-homa 18080  df-arw 18081  df-ida 18109
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator