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Theorem homarel 16680
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 5224 . . . 4 (((Base‘𝐶) × (Base‘𝐶)) × V) ⊆ (V × V)
2 homahom.h . . . . . . 7 𝐻 = (Homa𝐶)
3 eqid 2621 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
42homarcl 16672 . . . . . . 7 (𝑓 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
52, 3, 4homaf 16674 . . . . . 6 (𝑓 ∈ (𝑋𝐻𝑌) → 𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
62, 3homarcl2 16679 . . . . . . 7 (𝑓 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
76simpld 475 . . . . . 6 (𝑓 ∈ (𝑋𝐻𝑌) → 𝑋 ∈ (Base‘𝐶))
86simprd 479 . . . . . 6 (𝑓 ∈ (𝑋𝐻𝑌) → 𝑌 ∈ (Base‘𝐶))
95, 7, 8fovrnd 6803 . . . . 5 (𝑓 ∈ (𝑋𝐻𝑌) → (𝑋𝐻𝑌) ∈ 𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
10 elelpwi 4169 . . . . 5 ((𝑓 ∈ (𝑋𝐻𝑌) ∧ (𝑋𝐻𝑌) ∈ 𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V)) → 𝑓 ∈ (((Base‘𝐶) × (Base‘𝐶)) × V))
119, 10mpdan 702 . . . 4 (𝑓 ∈ (𝑋𝐻𝑌) → 𝑓 ∈ (((Base‘𝐶) × (Base‘𝐶)) × V))
121, 11sseldi 3599 . . 3 (𝑓 ∈ (𝑋𝐻𝑌) → 𝑓 ∈ (V × V))
1312ssriv 3605 . 2 (𝑋𝐻𝑌) ⊆ (V × V)
14 df-rel 5119 . 2 (Rel (𝑋𝐻𝑌) ↔ (𝑋𝐻𝑌) ⊆ (V × V))
1513, 14mpbir 221 1 Rel (𝑋𝐻𝑌)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1482  wcel 1989  Vcvv 3198  wss 3572  𝒫 cpw 4156   × cxp 5110  Rel wrel 5117  cfv 5886  (class class class)co 6647  Basecbs 15851  Homachoma 16667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-homa 16670
This theorem is referenced by:  homahom  16683  homadm  16684  homacd  16685  homadmcd  16686
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