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

Theorem fnhomeqhomf 16937
Description: If the Hom-set operation is a function it is equal to the corresponding functionalized Hom-set operation. (Contributed by AV, 1-Mar-2020.)
Hypotheses
Ref Expression
homffval.f 𝐹 = (Homf𝐶)
homffval.b 𝐵 = (Base‘𝐶)
homffval.h 𝐻 = (Hom ‘𝐶)
Assertion
Ref Expression
fnhomeqhomf (𝐻 Fn (𝐵 × 𝐵) → 𝐹 = 𝐻)

Proof of Theorem fnhomeqhomf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnov 7257 . 2 (𝐻 Fn (𝐵 × 𝐵) ↔ 𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
2 homffval.f . . . 4 𝐹 = (Homf𝐶)
3 homffval.b . . . 4 𝐵 = (Base‘𝐶)
4 homffval.h . . . 4 𝐻 = (Hom ‘𝐶)
52, 3, 4homffval 16936 . . 3 𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
6 eqeq2 2832 . . 3 (𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) → (𝐹 = 𝐻𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))))
75, 6mpbiri 260 . 2 (𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) → 𝐹 = 𝐻)
81, 7sylbi 219 1 (𝐻 Fn (𝐵 × 𝐵) → 𝐹 = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537   × cxp 5527   Fn wfn 6324  cfv 6329  (class class class)co 7131  cmpo 7133  Basecbs 16459  Hom chom 16552  Homf chomf 16913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5240  ax-pr 5304  ax-un 7437
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3752  df-csb 3860  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-nul 4268  df-if 4442  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4813  df-iun 4895  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5434  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-ov 7134  df-oprab 7135  df-mpo 7136  df-1st 7665  df-2nd 7666  df-homf 16917
This theorem is referenced by:  estrchomfeqhom  17362  rngchomfeqhom  44370  rngchomffvalALTV  44396  ringchomfeqhom  44416
  Copyright terms: Public domain W3C validator