MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  homffval Structured version   Visualization version   GIF version

Theorem homffval 17530
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 6839 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3 homffval.b . . . . . 6 𝐵 = (Base‘𝐶)
42, 3eqtr4di 2795 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5 fveq2 6839 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
6 homffval.h . . . . . . 7 𝐻 = (Hom ‘𝐶)
75, 6eqtr4di 2795 . . . . . 6 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
87oveqd 7368 . . . . 5 (𝑐 = 𝐶 → (𝑥(Hom ‘𝑐)𝑦) = (𝑥𝐻𝑦))
94, 4, 8mpoeq123dv 7426 . . . 4 (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
10 df-homf 17510 . . . 4 Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)))
113fvexi 6853 . . . . 5 𝐵 ∈ V
1211, 11mpoex 8004 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) ∈ V
139, 10, 12fvmpt 6945 . . 3 (𝐶 ∈ V → (Homf𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
14 fvprc 6831 . . . 4 𝐶 ∈ V → (Homf𝐶) = ∅)
15 fvprc 6831 . . . . . . 7 𝐶 ∈ V → (Base‘𝐶) = ∅)
163, 15eqtrid 2789 . . . . . 6 𝐶 ∈ V → 𝐵 = ∅)
1716olcd 872 . . . . 5 𝐶 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
18 0mpo0 7434 . . . . 5 ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = ∅)
1917, 18syl 17 . . . 4 𝐶 ∈ V → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = ∅)
2014, 19eqtr4d 2780 . . 3 𝐶 ∈ V → (Homf𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
2113, 20pm2.61i 182 . 2 (Homf𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
221, 21eqtri 2765 1 𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 845   = wceq 1541  wcel 2106  Vcvv 3443  c0 4280  cfv 6493  (class class class)co 7351  cmpo 7353  Basecbs 17043  Hom chom 17104  Homf chomf 17506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-homf 17510
This theorem is referenced by:  fnhomeqhomf  17531  homfval  17532  homffn  17533  homfeq  17534  oppchomf  17562  reschomf  17675
  Copyright terms: Public domain W3C validator