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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 2798 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
4 | eqid 2798 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
5 | eqid 2798 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
6 | 1, 2, 3, 4, 5 | hofval 17494 | . 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 6658 | . . 3 ⊢ (Homf ‘𝐶) ∈ V | |
8 | fvex 6658 | . . . . 5 ⊢ (Base‘𝐶) ∈ V | |
9 | 8, 8 | xpex 7456 | . . . 4 ⊢ ((Base‘𝐶) × (Base‘𝐶)) ∈ V |
10 | 9, 9 | mpoex 7760 | . . 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 7681 | . 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 1538 ∈ wcel 2111 〈cop 4531 ↦ cmpt 5110 × cxp 5517 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 1st c1st 7669 2nd c2nd 7670 Basecbs 16475 Hom chom 16568 compcco 16569 Catccat 16927 Homf chomf 16929 HomFchof 17490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-hof 17492 |
This theorem is referenced by: hof1 17496 |
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