Theorem homa1 18104

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 5167	. . . 4 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 ↔ ⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌))
2		homahom.h	. . . . 5 ⊢ 𝐻 = (Homa‘𝐶)
3		eqid 2740	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
4	2	homarcl 18095	. . . . 5 ⊢ (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
5		eqid 2740	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6	2, 3	homarcl2 18102	. . . . . 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 18099	. . . 4 ⊢ (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))))
10	1, 9	sylbi 217	. . 3 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))))
11	10	ibi 267	. 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 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ⟨cop 4654   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  Hom chom 17322  Hom_achoma 18090

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-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-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-ov 7451  df-homa 18093

This theorem is referenced by:  homadm  18107  homacd  18108  homadmcd  18109