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Mirrors > Home > MPE Home > Th. List > hof2 | Structured version Visualization version GIF version |
Description: The morphism part of the Hom functor, for morphisms 〈𝑓, 𝑔〉:〈𝑋, 𝑌〉⟶〈𝑍, 𝑊〉 (which since the first argument is contravariant means morphisms 𝑓:𝑍⟶𝑋 and 𝑔:𝑌⟶𝑊), yields a function (a morphism of SetCat) mapping ℎ:𝑋⟶𝑌 to 𝑔 ∘ ℎ ∘ 𝑓:𝑍⟶𝑊. (Contributed by Mario Carneiro, 15-Jan-2017.) |
Ref | Expression |
---|---|
hofval.m | ⊢ 𝑀 = (HomF‘𝐶) |
hofval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
hof1.b | ⊢ 𝐵 = (Base‘𝐶) |
hof1.h | ⊢ 𝐻 = (Hom ‘𝐶) |
hof1.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
hof1.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
hof2.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
hof2.w | ⊢ (𝜑 → 𝑊 ∈ 𝐵) |
hof2.o | ⊢ · = (comp‘𝐶) |
hof2.f | ⊢ (𝜑 → 𝐹 ∈ (𝑍𝐻𝑋)) |
hof2.g | ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑊)) |
hof2.k | ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) |
Ref | Expression |
---|---|
hof2 | ⊢ (𝜑 → ((𝐹(〈𝑋, 𝑌〉(2nd ‘𝑀)〈𝑍, 𝑊〉)𝐺)‘𝐾) = ((𝐺(〈𝑋, 𝑌〉 · 𝑊)𝐾)(〈𝑍, 𝑋〉 · 𝑊)𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hofval.m | . . 3 ⊢ 𝑀 = (HomF‘𝐶) | |
2 | hofval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
3 | hof1.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
4 | hof1.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
5 | hof1.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | hof1.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | hof2.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
8 | hof2.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ 𝐵) | |
9 | hof2.o | . . 3 ⊢ · = (comp‘𝐶) | |
10 | hof2.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑍𝐻𝑋)) | |
11 | hof2.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑊)) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | hof2val 17505 | . 2 ⊢ (𝜑 → (𝐹(〈𝑋, 𝑌〉(2nd ‘𝑀)〈𝑍, 𝑊〉)𝐺) = (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝐺(〈𝑋, 𝑌〉 · 𝑊)ℎ)(〈𝑍, 𝑋〉 · 𝑊)𝐹))) |
13 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ℎ = 𝐾) | |
14 | 13 | oveq2d 7171 | . . 3 ⊢ ((𝜑 ∧ ℎ = 𝐾) → (𝐺(〈𝑋, 𝑌〉 · 𝑊)ℎ) = (𝐺(〈𝑋, 𝑌〉 · 𝑊)𝐾)) |
15 | 14 | oveq1d 7170 | . 2 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ((𝐺(〈𝑋, 𝑌〉 · 𝑊)ℎ)(〈𝑍, 𝑋〉 · 𝑊)𝐹) = ((𝐺(〈𝑋, 𝑌〉 · 𝑊)𝐾)(〈𝑍, 𝑋〉 · 𝑊)𝐹)) |
16 | hof2.k | . 2 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) | |
17 | ovexd 7190 | . 2 ⊢ (𝜑 → ((𝐺(〈𝑋, 𝑌〉 · 𝑊)𝐾)(〈𝑍, 𝑋〉 · 𝑊)𝐹) ∈ V) | |
18 | 12, 15, 16, 17 | fvmptd 6774 | 1 ⊢ (𝜑 → ((𝐹(〈𝑋, 𝑌〉(2nd ‘𝑀)〈𝑍, 𝑊〉)𝐺)‘𝐾) = ((𝐺(〈𝑋, 𝑌〉 · 𝑊)𝐾)(〈𝑍, 𝑋〉 · 𝑊)𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 〈cop 4572 ‘cfv 6354 (class class class)co 7155 2nd c2nd 7687 Basecbs 16482 Hom chom 16575 compcco 16576 Catccat 16934 HomFchof 17497 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-hof 17499 |
This theorem is referenced by: yon12 17514 yon2 17515 |
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