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Theorem homarel 18081
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 5701 . . . 4 (((Base‘𝐶) × (Base‘𝐶)) × V) ⊆ (V × V)
2 homahom.h . . . . . . 7 𝐻 = (Homa𝐶)
3 eqid 2737 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
42homarcl 18073 . . . . . . 7 (𝑓 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
52, 3, 4homaf 18075 . . . . . 6 (𝑓 ∈ (𝑋𝐻𝑌) → 𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
62, 3homarcl2 18080 . . . . . . 7 (𝑓 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
76simpld 494 . . . . . 6 (𝑓 ∈ (𝑋𝐻𝑌) → 𝑋 ∈ (Base‘𝐶))
86simprd 495 . . . . . 6 (𝑓 ∈ (𝑋𝐻𝑌) → 𝑌 ∈ (Base‘𝐶))
95, 7, 8fovcdmd 7605 . . . . 5 (𝑓 ∈ (𝑋𝐻𝑌) → (𝑋𝐻𝑌) ∈ 𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
10 elelpwi 4610 . . . . 5 ((𝑓 ∈ (𝑋𝐻𝑌) ∧ (𝑋𝐻𝑌) ∈ 𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V)) → 𝑓 ∈ (((Base‘𝐶) × (Base‘𝐶)) × V))
119, 10mpdan 687 . . . 4 (𝑓 ∈ (𝑋𝐻𝑌) → 𝑓 ∈ (((Base‘𝐶) × (Base‘𝐶)) × V))
121, 11sselid 3981 . . 3 (𝑓 ∈ (𝑋𝐻𝑌) → 𝑓 ∈ (V × V))
1312ssriv 3987 . 2 (𝑋𝐻𝑌) ⊆ (V × V)
14 df-rel 5692 . 2 (Rel (𝑋𝐻𝑌) ↔ (𝑋𝐻𝑌) ⊆ (V × V))
1513, 14mpbir 231 1 Rel (𝑋𝐻𝑌)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  Vcvv 3480  wss 3951  𝒫 cpw 4600   × cxp 5683  Rel wrel 5690  cfv 6561  (class class class)co 7431  Basecbs 17247  Homachoma 18068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-homa 18071
This theorem is referenced by:  homahom  18084  homadm  18085  homacd  18086  homadmcd  18087
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