Theorem fnhomeqhomf 17736

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 7564	. 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 17735	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))
6		eqeq2 2747	. . 3 ⊢ (𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) → (𝐹 = 𝐻 ↔ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))))
7	5, 6	mpbiri 258	. 2 ⊢ (𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) → 𝐹 = 𝐻)
8	1, 7	sylbi 217	1 ⊢ (𝐻 Fn (𝐵 × 𝐵) → 𝐹 = 𝐻)

Syntax hints:   → wi 4   = wceq 1537   × cxp 5687   Fn wfn 6558  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  Basecbs 17245  Hom chom 17309  Hom_f chomf 17711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-homf 17715

This theorem is referenced by:  estrchomfeqhom  18191  rngchomfeqhom  20642  ringchomfeqhom  20671  rngchomffvalALTV  48122