homarcl2 - Metamath Proof Explorer

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Theorem homarcl2 18210

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 6919	. . . 4 ⊢ (𝐹 ∈ (𝐻‘⟨𝑋, 𝑌⟩) → ⟨𝑋, 𝑌⟩ ∈ dom 𝐻)
2		df-ov 7423	. . . 4 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
3	1, 2	eleq2s 2878	. . 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 18203	. . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
7	4, 5, 6	homaf 18205	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))
8	7	fdmd 6720	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → dom 𝐻 = (𝐵 × 𝐵))
9	3, 8	eleqtrd 2862	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
10		opelxp 5686	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵) ↔ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
11	9, 10	sylib 221	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 Vcvv 3450 𝒫 cpw 4556 ⟨cop 4589 × cxp 5648 dom cdm 5650 ‘cfv 6538 (class class class)co 7420 Basecbs 17387 Hom_achoma 18198

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7423 df-homa 18201

This theorem is used by: homarel 18211 homa1 18212 homahom2 18213 homadm 18215 homacd 18216 arwdm 18222 arwcd 18223 coahom 18245 arwlid 18247 arwrid 18248 arwass 18249

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