Theorem homa1 17668

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 = (Homa‘𝐶)

Assertion

Ref	Expression
homa1	⊢ (𝑍(𝑋𝐻𝑌)𝐹 → 𝑍 = ⟨𝑋, 𝑌⟩)

Proof of Theorem homa1

Step	Hyp	Ref	Expression
1		df-br 5071	. . . 4 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 ↔ ⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌))
2		homahom.h	. . . . 5 ⊢ 𝐻 = (Homa‘𝐶)
3		eqid 2738	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
4	2	homarcl 17659	. . . . 5 ⊢ (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
5		eqid 2738	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6	2, 3	homarcl2 17666	. . . . . 6 ⊢ (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
7	6	simpld 494	. . . . 5 ⊢ (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → 𝑋 ∈ (Base‘𝐶))
8	6	simprd 495	. . . . 5 ⊢ (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → 𝑌 ∈ (Base‘𝐶))
9	2, 3, 4, 5, 7, 8	elhoma 17663	. . . 4 ⊢ (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))))
10	1, 9	sylbi 216	. . 3 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))))
11	10	ibi 266	. 2 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)))
12	11	simpld 494	1 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → 𝑍 = ⟨𝑋, 𝑌⟩)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ⟨cop 4564   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Hom chom 16899  Hom_achoma 17654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-homa 17657

This theorem is referenced by:  homadm  17671  homacd  17672  homadmcd  17673