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Theorem homarel 17038
 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 5358 . . . 4 (((Base‘𝐶) × (Base‘𝐶)) × V) ⊆ (V × V)
2 homahom.h . . . . . . 7 𝐻 = (Homa𝐶)
3 eqid 2825 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
42homarcl 17030 . . . . . . 7 (𝑓 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
52, 3, 4homaf 17032 . . . . . 6 (𝑓 ∈ (𝑋𝐻𝑌) → 𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
62, 3homarcl2 17037 . . . . . . 7 (𝑓 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
76simpld 490 . . . . . 6 (𝑓 ∈ (𝑋𝐻𝑌) → 𝑋 ∈ (Base‘𝐶))
86simprd 491 . . . . . 6 (𝑓 ∈ (𝑋𝐻𝑌) → 𝑌 ∈ (Base‘𝐶))
95, 7, 8fovrnd 7066 . . . . 5 (𝑓 ∈ (𝑋𝐻𝑌) → (𝑋𝐻𝑌) ∈ 𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
10 elelpwi 4391 . . . . 5 ((𝑓 ∈ (𝑋𝐻𝑌) ∧ (𝑋𝐻𝑌) ∈ 𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V)) → 𝑓 ∈ (((Base‘𝐶) × (Base‘𝐶)) × V))
119, 10mpdan 680 . . . 4 (𝑓 ∈ (𝑋𝐻𝑌) → 𝑓 ∈ (((Base‘𝐶) × (Base‘𝐶)) × V))
121, 11sseldi 3825 . . 3 (𝑓 ∈ (𝑋𝐻𝑌) → 𝑓 ∈ (V × V))
1312ssriv 3831 . 2 (𝑋𝐻𝑌) ⊆ (V × V)
14 df-rel 5349 . 2 (Rel (𝑋𝐻𝑌) ↔ (𝑋𝐻𝑌) ⊆ (V × V))
1513, 14mpbir 223 1 Rel (𝑋𝐻𝑌)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1658   ∈ wcel 2166  Vcvv 3414   ⊆ wss 3798  𝒫 cpw 4378   × cxp 5340  Rel wrel 5347  ‘cfv 6123  (class class class)co 6905  Basecbs 16222  Homachoma 17025 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-homa 17028 This theorem is referenced by:  homahom  17041  homadm  17042  homacd  17043  homadmcd  17044
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