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Mirrors > Home > MPE Home > Th. List > hof1fval | Structured version Visualization version GIF version |
Description: The object part of the Hom functor is the Homf operation, which is just a functionalized version of Hom. That is, it is a two argument function, which maps 𝑋, 𝑌 to the set of morphisms from 𝑋 to 𝑌. (Contributed by Mario Carneiro, 15-Jan-2017.) |
Ref | Expression |
---|---|
hofval.m | ⊢ 𝑀 = (HomF‘𝐶) |
hofval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
Ref | Expression |
---|---|
hof1fval | ⊢ (𝜑 → (1st ‘𝑀) = (Homf ‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hofval.m | . . 3 ⊢ 𝑀 = (HomF‘𝐶) | |
2 | hofval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
3 | eqid 2726 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
4 | eqid 2726 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
5 | eqid 2726 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
6 | 1, 2, 3, 4, 5 | hofval 18277 | . 2 ⊢ (𝜑 → 𝑀 = 〈(Homf ‘𝐶), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(〈(1st ‘𝑦), (1st ‘𝑥)〉(comp‘𝐶)(2nd ‘𝑦))𝑓))))〉) |
7 | fvex 6914 | . . 3 ⊢ (Homf ‘𝐶) ∈ V | |
8 | fvex 6914 | . . . . 5 ⊢ (Base‘𝐶) ∈ V | |
9 | 8, 8 | xpex 7761 | . . . 4 ⊢ ((Base‘𝐶) × (Base‘𝐶)) ∈ V |
10 | 9, 9 | mpoex 8093 | . . 3 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(〈(1st ‘𝑦), (1st ‘𝑥)〉(comp‘𝐶)(2nd ‘𝑦))𝑓)))) ∈ V |
11 | 7, 10 | op1std 8013 | . 2 ⊢ (𝑀 = 〈(Homf ‘𝐶), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(〈(1st ‘𝑦), (1st ‘𝑥)〉(comp‘𝐶)(2nd ‘𝑦))𝑓))))〉 → (1st ‘𝑀) = (Homf ‘𝐶)) |
12 | 6, 11 | syl 17 | 1 ⊢ (𝜑 → (1st ‘𝑀) = (Homf ‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 〈cop 4639 ↦ cmpt 5236 × cxp 5680 ‘cfv 6554 (class class class)co 7424 ∈ cmpo 7426 1st c1st 8001 2nd c2nd 8002 Basecbs 17213 Hom chom 17277 compcco 17278 Catccat 17677 Homf chomf 17679 HomFchof 18273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-hof 18275 |
This theorem is referenced by: hof1 18279 |
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