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Theorem elhomai 16506
 Description: Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
homarcl.h 𝐻 = (Homa𝐶)
homafval.b 𝐵 = (Base‘𝐶)
homafval.c (𝜑𝐶 ∈ Cat)
homaval.j 𝐽 = (Hom ‘𝐶)
homaval.x (𝜑𝑋𝐵)
homaval.y (𝜑𝑌𝐵)
elhomai.f (𝜑𝐹 ∈ (𝑋𝐽𝑌))
Assertion
Ref Expression
elhomai (𝜑 → ⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹)

Proof of Theorem elhomai
StepHypRef Expression
1 eqidd 2611 . 2 (𝜑 → ⟨𝑋, 𝑌⟩ = ⟨𝑋, 𝑌⟩)
2 elhomai.f . 2 (𝜑𝐹 ∈ (𝑋𝐽𝑌))
3 homarcl.h . . 3 𝐻 = (Homa𝐶)
4 homafval.b . . 3 𝐵 = (Base‘𝐶)
5 homafval.c . . 3 (𝜑𝐶 ∈ Cat)
6 homaval.j . . 3 𝐽 = (Hom ‘𝐶)
7 homaval.x . . 3 (𝜑𝑋𝐵)
8 homaval.y . . 3 (𝜑𝑌𝐵)
93, 4, 5, 6, 7, 8elhoma 16505 . 2 (𝜑 → (⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹 ↔ (⟨𝑋, 𝑌⟩ = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌))))
101, 2, 9mpbir2and 959 1 (𝜑 → ⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779  Catccat 16148  Homachoma 16496 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-homa 16499 This theorem is referenced by:  elhomai2  16507
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