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Mirrors > Home > MPE Home > Th. List > catcco | Structured version Visualization version GIF version |
Description: Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
catcbas.c | ⊢ 𝐶 = (CatCat‘𝑈) |
catcbas.b | ⊢ 𝐵 = (Base‘𝐶) |
catcbas.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
catcco.o | ⊢ · = (comp‘𝐶) |
catcco.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
catcco.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
catcco.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
catcco.f | ⊢ (𝜑 → 𝐹 ∈ (𝑋 Func 𝑌)) |
catcco.g | ⊢ (𝜑 → 𝐺 ∈ (𝑌 Func 𝑍)) |
Ref | Expression |
---|---|
catcco | ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘func 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | catcbas.c | . . . 4 ⊢ 𝐶 = (CatCat‘𝑈) | |
2 | catcbas.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | catcbas.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
4 | catcco.o | . . . 4 ⊢ · = (comp‘𝐶) | |
5 | 1, 2, 3, 4 | catccofval 16750 | . . 3 ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))) |
6 | simprl 794 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑣 = 〈𝑋, 𝑌〉) | |
7 | 6 | fveq2d 6195 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = (2nd ‘〈𝑋, 𝑌〉)) |
8 | catcco.x | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | catcco.y | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
10 | op2ndg 7181 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) | |
11 | 8, 9, 10 | syl2anc 693 | . . . . . . 7 ⊢ (𝜑 → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) |
12 | 11 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) |
13 | 7, 12 | eqtrd 2656 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌) |
14 | simprr 796 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍) | |
15 | 13, 14 | oveq12d 6668 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑣) Func 𝑧) = (𝑌 Func 𝑍)) |
16 | 6 | fveq2d 6195 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → ( Func ‘𝑣) = ( Func ‘〈𝑋, 𝑌〉)) |
17 | df-ov 6653 | . . . . 5 ⊢ (𝑋 Func 𝑌) = ( Func ‘〈𝑋, 𝑌〉) | |
18 | 16, 17 | syl6eqr 2674 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → ( Func ‘𝑣) = (𝑋 Func 𝑌)) |
19 | eqidd 2623 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑔 ∘func 𝑓) = (𝑔 ∘func 𝑓)) | |
20 | 15, 18, 19 | mpt2eq123dv 6717 | . . 3 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)) = (𝑔 ∈ (𝑌 Func 𝑍), 𝑓 ∈ (𝑋 Func 𝑌) ↦ (𝑔 ∘func 𝑓))) |
21 | opelxpi 5148 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) | |
22 | 8, 9, 21 | syl2anc 693 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
23 | catcco.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
24 | ovex 6678 | . . . . 5 ⊢ (𝑌 Func 𝑍) ∈ V | |
25 | ovex 6678 | . . . . 5 ⊢ (𝑋 Func 𝑌) ∈ V | |
26 | 24, 25 | mpt2ex 7247 | . . . 4 ⊢ (𝑔 ∈ (𝑌 Func 𝑍), 𝑓 ∈ (𝑋 Func 𝑌) ↦ (𝑔 ∘func 𝑓)) ∈ V |
27 | 26 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑔 ∈ (𝑌 Func 𝑍), 𝑓 ∈ (𝑋 Func 𝑌) ↦ (𝑔 ∘func 𝑓)) ∈ V) |
28 | 5, 20, 22, 23, 27 | ovmpt2d 6788 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉 · 𝑍) = (𝑔 ∈ (𝑌 Func 𝑍), 𝑓 ∈ (𝑋 Func 𝑌) ↦ (𝑔 ∘func 𝑓))) |
29 | oveq12 6659 | . . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → (𝑔 ∘func 𝑓) = (𝐺 ∘func 𝐹)) | |
30 | 29 | adantl 482 | . 2 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝑔 ∘func 𝑓) = (𝐺 ∘func 𝐹)) |
31 | catcco.g | . 2 ⊢ (𝜑 → 𝐺 ∈ (𝑌 Func 𝑍)) | |
32 | catcco.f | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋 Func 𝑌)) | |
33 | ovexd 6680 | . 2 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ V) | |
34 | 28, 30, 31, 32, 33 | ovmpt2d 6788 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘func 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 〈cop 4183 × cxp 5112 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 2nd c2nd 7167 Basecbs 15857 compcco 15953 Func cfunc 16514 ∘func ccofu 16516 CatCatccatc 16744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-hom 15966 df-cco 15967 df-catc 16745 |
This theorem is referenced by: catccatid 16752 resscatc 16755 catcisolem 16756 catciso 16757 |
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