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 = (Homf ‘𝐶)
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.

Step	Hyp	Ref	Expression
1		fnov 7002	. 2 ⊢ (𝐻 Fn (𝐵 × 𝐵) ↔ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)))
2		homffval.f	. . . 4 ⊢ 𝐹 = (Homf ‘𝐶)
3		homffval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
4		homffval.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
5	2, 3, 4	homffval 16664	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))
6		eqeq2 2810	. . 3 ⊢ (𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) → (𝐹 = 𝐻 ↔ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))))
7	5, 6	mpbiri 250	. 2 ⊢ (𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) → 𝐹 = 𝐻)
8	1, 7	sylbi 209	1 ⊢ (𝐻 Fn (𝐵 × 𝐵) → 𝐹 = 𝐻)

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  Hom_f 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