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Theorem homa1 16734
 Description: The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypothesis
Ref Expression
homahom.h 𝐻 = (Homa𝐶)
Assertion
Ref Expression
homa1 (𝑍(𝑋𝐻𝑌)𝐹𝑍 = ⟨𝑋, 𝑌⟩)

Proof of Theorem homa1
StepHypRef Expression
1 df-br 4686 . . . 4 (𝑍(𝑋𝐻𝑌)𝐹 ↔ ⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌))
2 homahom.h . . . . 5 𝐻 = (Homa𝐶)
3 eqid 2651 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
42homarcl 16725 . . . . 5 (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
5 eqid 2651 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
62, 3homarcl2 16732 . . . . . 6 (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
76simpld 474 . . . . 5 (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → 𝑋 ∈ (Base‘𝐶))
86simprd 478 . . . . 5 (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → 𝑌 ∈ (Base‘𝐶))
92, 3, 4, 5, 7, 8elhoma 16729 . . . 4 (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))))
101, 9sylbi 207 . . 3 (𝑍(𝑋𝐻𝑌)𝐹 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))))
1110ibi 256 . 2 (𝑍(𝑋𝐻𝑌)𝐹 → (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)))
1211simpld 474 1 (𝑍(𝑋𝐻𝑌)𝐹𝑍 = ⟨𝑋, 𝑌⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ⟨cop 4216   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690  Basecbs 15904  Hom chom 15999  Homachoma 16720 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-homa 16723 This theorem is referenced by:  homadm  16737  homacd  16738  homadmcd  16739
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