Theorem elhomai 18078

Description: Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
homarcl.h	⊢ 𝐻 = (Homa‘𝐶)
homafval.b	⊢ 𝐵 = (Base‘𝐶)
homafval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
homaval.j	⊢ 𝐽 = (Hom ‘𝐶)
homaval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
homaval.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
elhomai.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐽𝑌))

Assertion

Ref	Expression
elhomai	⊢ (𝜑 → ⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹)

Proof of Theorem elhomai

Step	Hyp	Ref	Expression
1		eqidd 2738	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ = ⟨𝑋, 𝑌⟩)
2		elhomai.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐽𝑌))
3		homarcl.h	. . 3 ⊢ 𝐻 = (Homa‘𝐶)
4		homafval.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
5		homafval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
6		homaval.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐶)
7		homaval.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		homaval.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	3, 4, 5, 6, 7, 8	elhoma 18077	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹 ↔ (⟨𝑋, 𝑌⟩ = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌))))
10	1, 2, 9	mpbir2and 713	1 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ⟨cop 4632   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  Catccat 17707  Hom_achoma 18068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-homa 18071

This theorem is referenced by:  elhomai2  18079