homarcl2 - Metamath Proof Explorer

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Theorem homarcl2 18080

Description: Reverse closure for the domain and codomain of an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)

**Hypotheses**
Ref	Expression
homahom.h	⊢ 𝐻 = (Hom_a‘𝐶)
homarcl2.b	⊢ 𝐵 = (Base‘𝐶)

**Assertion**
Ref	Expression
homarcl2	⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))

**Proof of Theorem homarcl2**
Step	Hyp	Ref	Expression
1		elfvdm 6943	. . . 4 ⊢ (𝐹 ∈ (𝐻‘⟨𝑋, 𝑌⟩) → ⟨𝑋, 𝑌⟩ ∈ dom 𝐻)
2		df-ov 7434	. . . 4 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
3	1, 2	eleq2s 2859	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ⟨𝑋, 𝑌⟩ ∈ dom 𝐻)
4		homahom.h	. . . . 5 ⊢ 𝐻 = (Hom_a‘𝐶)
5		homarcl2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
6	4	homarcl 18073	. . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
7	4, 5, 6	homaf 18075	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))
8	7	fdmd 6746	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → dom 𝐻 = (𝐵 × 𝐵))
9	3, 8	eleqtrd 2843	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
10		opelxp 5721	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵) ↔ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
11	9, 10	sylib 218	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 𝒫 cpw 4600 ⟨cop 4632 × cxp 5683 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 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: homarel 18081 homa1 18082 homahom2 18083 homadm 18085 homacd 18086 arwdm 18092 arwcd 18093 coahom 18115 arwlid 18117 arwrid 18118 arwass 18119

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