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Theorem homahom2 18000

Description: The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)

**Hypotheses**
Ref	Expression
homahom.h	⊢ 𝐻 = (Hom_a‘𝐶)
homahom.j	⊢ 𝐽 = (Hom ‘𝐶)

**Assertion**
Ref	Expression
homahom2	⊢ (𝑍(𝑋𝐻𝑌)𝐹 → 𝐹 ∈ (𝑋𝐽𝑌))

**Proof of Theorem homahom2**
Step	Hyp	Ref	Expression
1		df-br 5076	. . . 4 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 ↔ ⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌))
2		homahom.h	. . . . 5 ⊢ 𝐻 = (Hom_a‘𝐶)
3		eqid 2741	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
4	2	homarcl 17990	. . . . 5 ⊢ (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
5		homahom.j	. . . . 5 ⊢ 𝐽 = (Hom ‘𝐶)
6	2, 3	homarcl2 17997	. . . . . 6 ⊢ (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
7	6	simpld 496	. . . . 5 ⊢ (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → 𝑋 ∈ (Base‘𝐶))
8	6	simprd 497	. . . . 5 ⊢ (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → 𝑌 ∈ (Base‘𝐶))
9	2, 3, 4, 5, 7, 8	elhoma 17994	. . . 4 ⊢ (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌))))
10	1, 9	sylbi 219	. . 3 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌))))
11	10	ibi 269	. 2 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌)))
12	11	simprd 497	1 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → 𝐹 ∈ (𝑋𝐽𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ⟨cop 4564 class class class wbr 5075 ‘cfv 6489 (class class class)co 7360 Basecbs 17174 Hom chom 17226 Hom_achoma 17985

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-ral 3056 df-rex 3066 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-ov 7363 df-homa 17988

This theorem is referenced by: homahom 18001

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