Theorem homarel 17290

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 5565	. . . 4 ⊢ (((Base‘𝐶) × (Base‘𝐶)) × V) ⊆ (V × V)
2		homahom.h	. . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶)
3		eqid 2821	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
4	2	homarcl 17282	. . . . . . 7 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
5	2, 3, 4	homaf 17284	. . . . . 6 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
6	2, 3	homarcl2 17289	. . . . . . 7 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
7	6	simpld 497	. . . . . 6 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝑋 ∈ (Base‘𝐶))
8	6	simprd 498	. . . . . 6 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝑌 ∈ (Base‘𝐶))
9	5, 7, 8	fovrnd 7314	. . . . 5 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → (𝑋𝐻𝑌) ∈ 𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
10		elelpwi 4553	. . . . 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	sseldi 3964	. . 3 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) → 𝑓 ∈ (V × V))
13	12	ssriv 3970	. 2 ⊢ (𝑋𝐻𝑌) ⊆ (V × V)
14		df-rel 5556	. 2 ⊢ (Rel (𝑋𝐻𝑌) ↔ (𝑋𝐻𝑌) ⊆ (V × V))
15	13, 14	mpbir 233	1 ⊢ Rel (𝑋𝐻𝑌)

Syntax hints:   = wceq 1533   ∈ wcel 2110  Vcvv 3494   ⊆ wss 3935  𝒫 cpw 4538   × cxp 5547  Rel wrel 5554  ‘cfv 6349  (class class class)co 7150  Basecbs 16477  Hom_achoma 17277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-homa 17280

This theorem is referenced by:  homahom  17293  homadm  17294  homacd  17295  homadmcd  17296