Theorem fnhomeqhomf 17664

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 7546	. 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 17663	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))
6		eqeq2 2739	. . 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 216	1 ⊢ (𝐻 Fn (𝐵 × 𝐵) → 𝐹 = 𝐻)

Syntax hints:   → wi 4   = wceq 1534   × cxp 5670   Fn wfn 6537  ‘cfv 6542  (class class class)co 7414   ∈ cmpo 7416  Basecbs 17173  Hom chom 17237  Hom_f chomf 17639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-homf 17643

This theorem is referenced by:  estrchomfeqhom  18119  rngchomfeqhom  20551  ringchomfeqhom  20580  rngchomffvalALTV  47312