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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 17508 | . 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 776 | . . . . 5 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ ℎ ∈ (𝑋𝐻𝑌)) → 𝑔 = 𝐺) | |
12 | 11 | oveq1d 7174 | . . . 4 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ ℎ ∈ (𝑋𝐻𝑌)) → (𝑔(〈𝑋, 𝑌〉 · 𝑊)ℎ) = (𝐺(〈𝑋, 𝑌〉 · 𝑊)ℎ)) |
13 | simplrl 775 | . . . 4 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ ℎ ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝐹) | |
14 | 12, 13 | oveq12d 7177 | . . 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 5164 | . 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 7192 | . . . 4 ⊢ (𝑋𝐻𝑌) ∈ V | |
19 | 18 | mptex 6989 | . . 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 7305 | 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 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 〈cop 4576 ↦ cmpt 5149 ‘cfv 6358 (class class class)co 7159 2nd c2nd 7691 Basecbs 16486 Hom chom 16579 compcco 16580 Catccat 16938 HomFchof 17501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-hof 17503 |
This theorem is referenced by: hof2 17510 hofcllem 17511 hofcl 17512 yonedalem3b 17532 |
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