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Theorem fnhomeqhomf 16665
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 7002 . 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 16664 . . 3 𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
6 eqeq2 2810 . . 3 (𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) → (𝐹 = 𝐻𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))))
75, 6mpbiri 250 . 2 (𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) → 𝐹 = 𝐻)
81, 7sylbi 209 1 (𝐻 Fn (𝐵 × 𝐵) → 𝐹 = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653   × cxp 5310   Fn wfn 6096  cfv 6101  (class class class)co 6878  cmpt2 6880  Basecbs 16184  Hom chom 16278  Homf chomf 16641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-homf 16645
This theorem is referenced by:  estrchomfeqhom  17090  rngchomfeqhom  42768  rngchomffvalALTV  42794  ringchomfeqhom  42814
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