Theorem elhomai2 17990

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
elhomai2	⊢ (𝜑 → ⟨𝑋, 𝑌, 𝐹⟩ ∈ (𝑋𝐻𝑌))

Proof of Theorem elhomai2

Step	Hyp	Ref	Expression
1		df-ot 4566	. 2 ⊢ ⟨𝑋, 𝑌, 𝐹⟩ = ⟨⟨𝑋, 𝑌⟩, 𝐹⟩
2		homarcl.h	. . . 4 ⊢ 𝐻 = (Homa‘𝐶)
3		homafval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
4		homafval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		homaval.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐶)
6		homaval.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		homaval.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
8		elhomai.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐽𝑌))
9	2, 3, 4, 5, 6, 7, 8	elhomai 17989	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹)
10		df-br 5075	. . 3 ⊢ (⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹 ↔ ⟨⟨𝑋, 𝑌⟩, 𝐹⟩ ∈ (𝑋𝐻𝑌))
11	9, 10	sylib 218	. 2 ⊢ (𝜑 → ⟨⟨𝑋, 𝑌⟩, 𝐹⟩ ∈ (𝑋𝐻𝑌))
12	1, 11	eqeltrid 2839	1 ⊢ (𝜑 → ⟨𝑋, 𝑌, 𝐹⟩ ∈ (𝑋𝐻𝑌))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ⟨cop 4563  ⟨cotp 4565   class class class wbr 5074  ‘cfv 6487  (class class class)co 7356  Basecbs 17168  Hom chom 17220  Catccat 17619  Hom_achoma 17979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-ot 4566  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-homa 17982

This theorem is referenced by:  idahom  18016  coahom  18026  termcarweu  49991  arweuthinc  49992  arweutermc  49993