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Theorem homarel 16733
Description: An arrow is an ordered pair. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypothesis
Ref Expression
homahom.h 𝐻 = (Homa𝐶)
Assertion
Ref Expression
homarel Rel (𝑋𝐻𝑌)

Proof of Theorem homarel
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 xpss 5159 . . . 4 (((Base‘𝐶) × (Base‘𝐶)) × V) ⊆ (V × V)
2 homahom.h . . . . . . 7 𝐻 = (Homa𝐶)
3 eqid 2651 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
42homarcl 16725 . . . . . . 7 (𝑓 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
52, 3, 4homaf 16727 . . . . . 6 (𝑓 ∈ (𝑋𝐻𝑌) → 𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
62, 3homarcl2 16732 . . . . . . 7 (𝑓 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
76simpld 474 . . . . . 6 (𝑓 ∈ (𝑋𝐻𝑌) → 𝑋 ∈ (Base‘𝐶))
86simprd 478 . . . . . 6 (𝑓 ∈ (𝑋𝐻𝑌) → 𝑌 ∈ (Base‘𝐶))
95, 7, 8fovrnd 6848 . . . . 5 (𝑓 ∈ (𝑋𝐻𝑌) → (𝑋𝐻𝑌) ∈ 𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
10 elelpwi 4204 . . . . 5 ((𝑓 ∈ (𝑋𝐻𝑌) ∧ (𝑋𝐻𝑌) ∈ 𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V)) → 𝑓 ∈ (((Base‘𝐶) × (Base‘𝐶)) × V))
119, 10mpdan 703 . . . 4 (𝑓 ∈ (𝑋𝐻𝑌) → 𝑓 ∈ (((Base‘𝐶) × (Base‘𝐶)) × V))
121, 11sseldi 3634 . . 3 (𝑓 ∈ (𝑋𝐻𝑌) → 𝑓 ∈ (V × V))
1312ssriv 3640 . 2 (𝑋𝐻𝑌) ⊆ (V × V)
14 df-rel 5150 . 2 (Rel (𝑋𝐻𝑌) ↔ (𝑋𝐻𝑌) ⊆ (V × V))
1513, 14mpbir 221 1 Rel (𝑋𝐻𝑌)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  Vcvv 3231  wss 3607  𝒫 cpw 4191   × cxp 5141  Rel wrel 5148  cfv 5926  (class class class)co 6690  Basecbs 15904  Homachoma 16720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-homa 16723
This theorem is referenced by:  homahom  16736  homadm  16737  homacd  16738  homadmcd  16739
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