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Theorem hof1 18240
Description: The object part of the Hom functor maps 𝑋, 𝑌 to the set of morphisms from 𝑋 to 𝑌. (Contributed by Mario Carneiro, 15-Jan-2017.)
Hypotheses
Ref Expression
hofval.m 𝑀 = (HomF𝐶)
hofval.c (𝜑𝐶 ∈ Cat)
hof1.b 𝐵 = (Base‘𝐶)
hof1.h 𝐻 = (Hom ‘𝐶)
hof1.x (𝜑𝑋𝐵)
hof1.y (𝜑𝑌𝐵)
Assertion
Ref Expression
hof1 (𝜑 → (𝑋(1st𝑀)𝑌) = (𝑋𝐻𝑌))

Proof of Theorem hof1
StepHypRef Expression
1 hofval.m . . . 4 𝑀 = (HomF𝐶)
2 hofval.c . . . 4 (𝜑𝐶 ∈ Cat)
31, 2hof1fval 18239 . . 3 (𝜑 → (1st𝑀) = (Homf𝐶))
43oveqd 7432 . 2 (𝜑 → (𝑋(1st𝑀)𝑌) = (𝑋(Homf𝐶)𝑌))
5 eqid 2728 . . 3 (Homf𝐶) = (Homf𝐶)
6 hof1.b . . 3 𝐵 = (Base‘𝐶)
7 hof1.h . . 3 𝐻 = (Hom ‘𝐶)
8 hof1.x . . 3 (𝜑𝑋𝐵)
9 hof1.y . . 3 (𝜑𝑌𝐵)
105, 6, 7, 8, 9homfval 17666 . 2 (𝜑 → (𝑋(Homf𝐶)𝑌) = (𝑋𝐻𝑌))
114, 10eqtrd 2768 1 (𝜑 → (𝑋(1st𝑀)𝑌) = (𝑋𝐻𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  cfv 6543  (class class class)co 7415  1st c1st 7986  Basecbs 17174  Hom chom 17238  Catccat 17638  Homf chomf 17640  HomFchof 18234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7988  df-2nd 7989  df-homf 17644  df-hof 18236
This theorem is referenced by:  yon11  18250  yonedalem21  18259
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