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Theorem fnhomeqhomf 16174
 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 6666 . 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 16173 . . 3 𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
6 eqeq2 2621 . . 3 (𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) → (𝐹 = 𝐻𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))))
75, 6mpbiri 247 . 2 (𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) → 𝐹 = 𝐻)
81, 7sylbi 206 1 (𝐻 Fn (𝐵 × 𝐵) → 𝐹 = 𝐻)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   × cxp 5036   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Basecbs 15695  Hom chom 15779  Homf chomf 16150 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-homf 16154 This theorem is referenced by:  estrchomfeqhom  16599  rngchomfeqhom  41761  rngchomffvalALTV  41787  ringchomfeqhom  41807
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