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Theorem homffval 17669
Description: Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)
Hypotheses
Ref Expression
homffval.f 𝐹 = (Homf𝐶)
homffval.b 𝐵 = (Base‘𝐶)
homffval.h 𝐻 = (Hom ‘𝐶)
Assertion
Ref Expression
homffval 𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐻,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem homffval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 homffval.f . 2 𝐹 = (Homf𝐶)
2 fveq2 6892 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3 homffval.b . . . . . 6 𝐵 = (Base‘𝐶)
42, 3eqtr4di 2783 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5 fveq2 6892 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
6 homffval.h . . . . . . 7 𝐻 = (Hom ‘𝐶)
75, 6eqtr4di 2783 . . . . . 6 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
87oveqd 7433 . . . . 5 (𝑐 = 𝐶 → (𝑥(Hom ‘𝑐)𝑦) = (𝑥𝐻𝑦))
94, 4, 8mpoeq123dv 7492 . . . 4 (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
10 df-homf 17649 . . . 4 Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)))
113fvexi 6906 . . . . 5 𝐵 ∈ V
1211, 11mpoex 8082 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) ∈ V
139, 10, 12fvmpt 7000 . . 3 (𝐶 ∈ V → (Homf𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
14 fvprc 6884 . . . 4 𝐶 ∈ V → (Homf𝐶) = ∅)
15 fvprc 6884 . . . . . . 7 𝐶 ∈ V → (Base‘𝐶) = ∅)
163, 15eqtrid 2777 . . . . . 6 𝐶 ∈ V → 𝐵 = ∅)
1716olcd 872 . . . . 5 𝐶 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
18 0mpo0 7500 . . . . 5 ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = ∅)
1917, 18syl 17 . . . 4 𝐶 ∈ V → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = ∅)
2014, 19eqtr4d 2768 . . 3 𝐶 ∈ V → (Homf𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
2113, 20pm2.61i 182 . 2 (Homf𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
221, 21eqtri 2753 1 𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 845   = wceq 1533  wcel 2098  Vcvv 3463  c0 4318  cfv 6543  (class class class)co 7416  cmpo 7418  Basecbs 17179  Hom chom 17243  Homf chomf 17645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992  df-homf 17649
This theorem is referenced by:  fnhomeqhomf  17670  homfval  17671  homffn  17672  homfeq  17673  oppchomf  17701  reschomf  17814
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