homarcl2 - Metamath Proof Explorer

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Theorem homarcl2 17991

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 6866	. . . 4 ⊢ (𝐹 ∈ (𝐻‘⟨𝑋, 𝑌⟩) → ⟨𝑋, 𝑌⟩ ∈ dom 𝐻)
2		df-ov 7361	. . . 4 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
3	1, 2	eleq2s 2855	. . 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 17984	. . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
7	4, 5, 6	homaf 17986	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))
8	7	fdmd 6670	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → dom 𝐻 = (𝐵 × 𝐵))
9	3, 8	eleqtrd 2839	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
10		opelxp 5658	. 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 1542 ∈ wcel 2114 Vcvv 3430 𝒫 cpw 4542 ⟨cop 4574 × cxp 5620 dom cdm 5622 ‘cfv 6490 (class class class)co 7358 Basecbs 17168 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 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-ov 7361 df-homa 17982

This theorem is referenced by: homarel 17992 homa1 17993 homahom2 17994 homadm 17996 homacd 17997 arwdm 18003 arwcd 18004 coahom 18026 arwlid 18028 arwrid 18029 arwass 18030

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