fnhomeqhomf - Metamath Proof Explorer

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Theorem fnhomeqhomf 17894

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 = (Hom_f ‘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 7574	. 2 ⊢ (𝐻 Fn (𝐵 × 𝐵) ↔ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)))
2		homffval.f	. . . 4 ⊢ 𝐹 = (Hom_f ‘𝐶)
3		homffval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
4		homffval.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
5	2, 3, 4	homffval 17893	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))
6		eqeq2 2775	. . 3 ⊢ (𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) → (𝐹 = 𝐻 ↔ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))))
7	5, 6	mpbiri 261	. 2 ⊢ (𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) → 𝐹 = 𝐻)
8	1, 7	sylbi 220	1 ⊢ (𝐻 Fn (𝐵 × 𝐵) → 𝐹 = 𝐻)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 × cxp 5653 Fn wfn 6553 ‘cfv 6558 (class class class)co 7443 ∈ cmpo 7445 Basecbs 17416 Hom chom 17468 Hom_f chomf 17869

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-ov 7446 df-oprab 7447 df-mpo 7448 df-1st 8025 df-2nd 8026 df-homf 17873

This theorem is used by: estrchomfeqhom 18339 rngchomfeqhom 20907 ringchomfeqhom 20936 rngchomffvalALTV 49580

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