Theorem homarel 18003

Description: An arrow is an ordered pair. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypothesis

Ref	Expression
homahom.h	⊢ 𝐻 = (Homa‘𝐶)

Assertion

Ref	Expression
homarel	⊢ Rel (𝑋𝐻𝑌)

Proof of Theorem homarel

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpss 5647	. . . 4 ⊢ (((Base‘𝐶) × (Base‘𝐶)) × V) ⊆ (V × V)
2		homahom.h	. . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶)
3		eqid 2736	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
4	2	homarcl 17995	. . . . . . 7 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
5	2, 3, 4	homaf 17997	. . . . . 6 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
6	2, 3	homarcl2 18002	. . . . . . 7 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
7	6	simpld 494	. . . . . 6 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝑋 ∈ (Base‘𝐶))
8	6	simprd 495	. . . . . 6 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝑌 ∈ (Base‘𝐶))
9	5, 7, 8	fovcdmd 7539	. . . . 5 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → (𝑋𝐻𝑌) ∈ 𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
10		elelpwi 4551	. . . . 5 ⊢ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ (𝑋𝐻𝑌) ∈ 𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V)) → 𝑓 ∈ (((Base‘𝐶) × (Base‘𝐶)) × V))
11	9, 10	mpdan 688	. . . 4 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝑓 ∈ (((Base‘𝐶) × (Base‘𝐶)) × V))
12	1, 11	sselid 3919	. . 3 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝑓 ∈ (V × V))
13	12	ssriv 3925	. 2 ⊢ (𝑋𝐻𝑌) ⊆ (V × V)
14		df-rel 5638	. 2 ⊢ (Rel (𝑋𝐻𝑌) ↔ (𝑋𝐻𝑌) ⊆ (V × V))
15	13, 14	mpbir 231	1 ⊢ Rel (𝑋𝐻𝑌)

Syntax hints:   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ⊆ wss 3889  𝒫 cpw 4541   × cxp 5629  Rel wrel 5636  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  Hom_achoma 17990

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-homa 17993

This theorem is referenced by:  homahom  18006  homadm  18007  homacd  18008  homadmcd  18009