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Mirrors > Home > MPE Home > Th. List > hof1 | Structured version Visualization version GIF version |
Description: The object part of the Hom functor 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) |
hof1.b | ⊢ 𝐵 = (Base‘𝐶) |
hof1.h | ⊢ 𝐻 = (Hom ‘𝐶) |
hof1.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
hof1.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
hof1 | ⊢ (𝜑 → (𝑋(1st ‘𝑀)𝑌) = (𝑋𝐻𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hofval.m | . . . 4 ⊢ 𝑀 = (HomF‘𝐶) | |
2 | hofval.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
3 | 1, 2 | hof1fval 18202 | . . 3 ⊢ (𝜑 → (1st ‘𝑀) = (Homf ‘𝐶)) |
4 | 3 | oveqd 7422 | . 2 ⊢ (𝜑 → (𝑋(1st ‘𝑀)𝑌) = (𝑋(Homf ‘𝐶)𝑌)) |
5 | eqid 2732 | . . 3 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
6 | hof1.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
7 | hof1.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
8 | hof1.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | hof1.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
10 | 5, 6, 7, 8, 9 | homfval 17632 | . 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐶)𝑌) = (𝑋𝐻𝑌)) |
11 | 4, 10 | eqtrd 2772 | 1 ⊢ (𝜑 → (𝑋(1st ‘𝑀)𝑌) = (𝑋𝐻𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ‘cfv 6540 (class class class)co 7405 1st c1st 7969 Basecbs 17140 Hom chom 17204 Catccat 17604 Homf chomf 17606 HomFchof 18197 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-homf 17610 df-hof 18199 |
This theorem is referenced by: yon11 18213 yonedalem21 18222 |
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