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Mirrors > Home > MPE Home > Th. List > hof2val | 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)) |
Ref | Expression |
---|---|
hof2val | ⊢ (𝜑 → (𝐹(〈𝑋, 𝑌〉(2nd ‘𝑀)〈𝑍, 𝑊〉)𝐺) = (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝐺(〈𝑋, 𝑌〉 · 𝑊)ℎ)(〈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 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | hof2fval 17493 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉(2nd ‘𝑀)〈𝑍, 𝑊〉) = (𝑓 ∈ (𝑍𝐻𝑋), 𝑔 ∈ (𝑌𝐻𝑊) ↦ (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝑔(〈𝑋, 𝑌〉 · 𝑊)ℎ)(〈𝑍, 𝑋〉 · 𝑊)𝑓)))) |
11 | simplrr 774 | . . . . 5 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ ℎ ∈ (𝑋𝐻𝑌)) → 𝑔 = 𝐺) | |
12 | 11 | oveq1d 7160 | . . . 4 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ ℎ ∈ (𝑋𝐻𝑌)) → (𝑔(〈𝑋, 𝑌〉 · 𝑊)ℎ) = (𝐺(〈𝑋, 𝑌〉 · 𝑊)ℎ)) |
13 | simplrl 773 | . . . 4 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ ℎ ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝐹) | |
14 | 12, 13 | oveq12d 7163 | . . 3 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ ℎ ∈ (𝑋𝐻𝑌)) → ((𝑔(〈𝑋, 𝑌〉 · 𝑊)ℎ)(〈𝑍, 𝑋〉 · 𝑊)𝑓) = ((𝐺(〈𝑋, 𝑌〉 · 𝑊)ℎ)(〈𝑍, 𝑋〉 · 𝑊)𝐹)) |
15 | 14 | mpteq2dva 5152 | . 2 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝑔(〈𝑋, 𝑌〉 · 𝑊)ℎ)(〈𝑍, 𝑋〉 · 𝑊)𝑓)) = (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝐺(〈𝑋, 𝑌〉 · 𝑊)ℎ)(〈𝑍, 𝑋〉 · 𝑊)𝐹))) |
16 | hof2.f | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑍𝐻𝑋)) | |
17 | hof2.g | . 2 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑊)) | |
18 | ovex 7178 | . . . 4 ⊢ (𝑋𝐻𝑌) ∈ V | |
19 | 18 | mptex 6977 | . . 3 ⊢ (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝐺(〈𝑋, 𝑌〉 · 𝑊)ℎ)(〈𝑍, 𝑋〉 · 𝑊)𝐹)) ∈ V |
20 | 19 | a1i 11 | . 2 ⊢ (𝜑 → (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝐺(〈𝑋, 𝑌〉 · 𝑊)ℎ)(〈𝑍, 𝑋〉 · 𝑊)𝐹)) ∈ V) |
21 | 10, 15, 16, 17, 20 | ovmpod 7291 | 1 ⊢ (𝜑 → (𝐹(〈𝑋, 𝑌〉(2nd ‘𝑀)〈𝑍, 𝑊〉)𝐺) = (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝐺(〈𝑋, 𝑌〉 · 𝑊)ℎ)(〈𝑍, 𝑋〉 · 𝑊)𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 〈cop 4563 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 2nd c2nd 7677 Basecbs 16471 Hom chom 16564 compcco 16565 Catccat 16923 HomFchof 17486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-hof 17488 |
This theorem is referenced by: hof2 17495 hofcllem 17496 hofcl 17497 yonedalem3b 17517 |
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