homarcl2 - Metamath Proof Explorer

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Theorem homarcl2 18049

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 6895	. . . 4 ⊢ (𝐹 ∈ (𝐻‘⟨𝑋, 𝑌⟩) → ⟨𝑋, 𝑌⟩ ∈ dom 𝐻)
2		df-ov 7393	. . . 4 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
3	1, 2	eleq2s 2879	. . 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 18042	. . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
7	4, 5, 6	homaf 18044	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))
8	7	fdmd 6696	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → dom 𝐻 = (𝐵 × 𝐵))
9	3, 8	eleqtrd 2863	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
10		opelxp 5681	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵) ↔ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
11	9, 10	sylib 220	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 Vcvv 3453 𝒫 cpw 4554 ⟨cop 4587 × cxp 5643 dom cdm 5645 ‘cfv 6515 (class class class)co 7390 Basecbs 17226 Hom_achoma 18037

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7712

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5540 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-ov 7393 df-homa 18040

This theorem is referenced by: homarel 18050 homa1 18051 homahom2 18052 homadm 18054 homacd 18055 arwdm 18061 arwcd 18062 coahom 18084 arwlid 18086 arwrid 18087 arwass 18088

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