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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 16719 | . 2 ⊢ (𝜑 → (𝐹(〈𝑋, 𝑌〉(2nd ‘𝑀)〈𝑍, 𝑊〉)𝐺) = (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝐺(〈𝑋, 𝑌〉 · 𝑊)ℎ)(〈𝑍, 𝑋〉 · 𝑊)𝐹))) |
13 | simpr 476 | . . . 4 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ℎ = 𝐾) | |
14 | 13 | oveq2d 6565 | . . 3 ⊢ ((𝜑 ∧ ℎ = 𝐾) → (𝐺(〈𝑋, 𝑌〉 · 𝑊)ℎ) = (𝐺(〈𝑋, 𝑌〉 · 𝑊)𝐾)) |
15 | 14 | oveq1d 6564 | . 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 | ovex 6577 | . . 3 ⊢ ((𝐺(〈𝑋, 𝑌〉 · 𝑊)𝐾)(〈𝑍, 𝑋〉 · 𝑊)𝐹) ∈ V | |
18 | 17 | a1i 11 | . 2 ⊢ (𝜑 → ((𝐺(〈𝑋, 𝑌〉 · 𝑊)𝐾)(〈𝑍, 𝑋〉 · 𝑊)𝐹) ∈ V) |
19 | 12, 15, 16, 18 | fvmptd 6197 | 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 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 ‘cfv 5804 (class class class)co 6549 2nd c2nd 7058 Basecbs 15695 Hom chom 15779 compcco 15780 Catccat 16148 HomFchof 16711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-hof 16713 |
This theorem is referenced by: yon12 16728 yon2 16729 |
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