Theorem homarel 17985

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 5692	. . . 4 ⊢ (((Base‘𝐶) × (Base‘𝐶)) × V) ⊆ (V × V)
2		homahom.h	. . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶)
3		eqid 2732	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
4	2	homarcl 17977	. . . . . . 7 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
5	2, 3, 4	homaf 17979	. . . . . 6 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
6	2, 3	homarcl2 17984	. . . . . . 7 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
7	6	simpld 495	. . . . . 6 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝑋 ∈ (Base‘𝐶))
8	6	simprd 496	. . . . . 6 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝑌 ∈ (Base‘𝐶))
9	5, 7, 8	fovcdmd 7578	. . . . 5 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → (𝑋𝐻𝑌) ∈ 𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
10		elelpwi 4612	. . . . 5 ⊢ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ (𝑋𝐻𝑌) ∈ 𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V)) → 𝑓 ∈ (((Base‘𝐶) × (Base‘𝐶)) × V))
11	9, 10	mpdan 685	. . . 4 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝑓 ∈ (((Base‘𝐶) × (Base‘𝐶)) × V))
12	1, 11	sselid 3980	. . 3 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝑓 ∈ (V × V))
13	12	ssriv 3986	. 2 ⊢ (𝑋𝐻𝑌) ⊆ (V × V)
14		df-rel 5683	. 2 ⊢ (Rel (𝑋𝐻𝑌) ↔ (𝑋𝐻𝑌) ⊆ (V × V))
15	13, 14	mpbir 230	1 ⊢ Rel (𝑋𝐻𝑌)

Syntax hints:   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ⊆ wss 3948  𝒫 cpw 4602   × cxp 5674  Rel wrel 5681  ‘cfv 6543  (class class class)co 7408  Basecbs 17143  Hom_achoma 17972

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-homa 17975

This theorem is referenced by:  homahom  17988  homadm  17989  homacd  17990  homadmcd  17991