homarcl2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > homarcl2		Structured version Visualization version GIF version

Theorem homarcl2 18004

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 6898	. . . 4 ⊢ (𝐹 ∈ (𝐻‘⟨𝑋, 𝑌⟩) → ⟨𝑋, 𝑌⟩ ∈ dom 𝐻)
2		df-ov 7393	. . . 4 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
3	1, 2	eleq2s 2847	. . 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 17997	. . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
7	4, 5, 6	homaf 17999	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))
8	7	fdmd 6701	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → dom 𝐻 = (𝐵 × 𝐵))
9	3, 8	eleqtrd 2831	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
10		opelxp 5677	. 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 2109 Vcvv 3450 𝒫 cpw 4566 ⟨cop 4598 × cxp 5639 dom cdm 5641 ‘cfv 6514 (class class class)co 7390 Basecbs 17186 Hom_achoma 17992

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-ov 7393 df-homa 17995

This theorem is referenced by: homarel 18005 homa1 18006 homahom2 18007 homadm 18009 homacd 18010 arwdm 18016 arwcd 18017 coahom 18039 arwlid 18041 arwrid 18042 arwass 18043

W3C validator