Theorem homa1 18090

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 5148	. . . 4 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 ↔ ⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌))
2		homahom.h	. . . . 5 ⊢ 𝐻 = (Homa‘𝐶)
3		eqid 2734	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
4	2	homarcl 18081	. . . . 5 ⊢ (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
5		eqid 2734	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6	2, 3	homarcl2 18088	. . . . . 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 18085	. . . 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 1536   ∈ wcel 2105  ⟨cop 4636   class class class wbr 5147  ‘cfv 6562  (class class class)co 7430  Basecbs 17244  Hom chom 17308  Hom_achoma 18076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-homa 18079

This theorem is referenced by:  homadm  18093  homacd  18094  homadmcd  18095