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Theorem homarel 18089
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 5704 . . . 4 (((Base‘𝐶) × (Base‘𝐶)) × V) ⊆ (V × V)
2 homahom.h . . . . . . 7 𝐻 = (Homa𝐶)
3 eqid 2734 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
42homarcl 18081 . . . . . . 7 (𝑓 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
52, 3, 4homaf 18083 . . . . . 6 (𝑓 ∈ (𝑋𝐻𝑌) → 𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
62, 3homarcl2 18088 . . . . . . 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 7604 . . . . 5 (𝑓 ∈ (𝑋𝐻𝑌) → (𝑋𝐻𝑌) ∈ 𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
10 elelpwi 4614 . . . . 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 3992 . . 3 (𝑓 ∈ (𝑋𝐻𝑌) → 𝑓 ∈ (V × V))
1312ssriv 3998 . 2 (𝑋𝐻𝑌) ⊆ (V × V)
14 df-rel 5695 . 2 (Rel (𝑋𝐻𝑌) ↔ (𝑋𝐻𝑌) ⊆ (V × V))
1513, 14mpbir 231 1 Rel (𝑋𝐻𝑌)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536  wcel 2105  Vcvv 3477  wss 3962  𝒫 cpw 4604   × cxp 5686  Rel wrel 5693  cfv 6562  (class class class)co 7430  Basecbs 17244  Homachoma 18076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-homa 18079
This theorem is referenced by:  homahom  18092  homadm  18093  homacd  18094  homadmcd  18095
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