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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 18149 | . . 3 ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))) | 
| 6 | simprl 771 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑣 = 〈𝑋, 𝑌〉) | |
| 7 | 6 | fveq2d 6910 | . . . . . 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 8027 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) | |
| 11 | 8, 9, 10 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) | 
| 12 | 11 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) | 
| 13 | 7, 12 | eqtrd 2777 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌) | 
| 14 | simprr 773 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍) | |
| 15 | 13, 14 | oveq12d 7449 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑣) Func 𝑧) = (𝑌 Func 𝑍)) | 
| 16 | 6 | fveq2d 6910 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → ( Func ‘𝑣) = ( Func ‘〈𝑋, 𝑌〉)) | 
| 17 | df-ov 7434 | . . . . 5 ⊢ (𝑋 Func 𝑌) = ( Func ‘〈𝑋, 𝑌〉) | |
| 18 | 16, 17 | eqtr4di 2795 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → ( Func ‘𝑣) = (𝑋 Func 𝑌)) | 
| 19 | eqidd 2738 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑔 ∘func 𝑓) = (𝑔 ∘func 𝑓)) | |
| 20 | 15, 18, 19 | mpoeq123dv 7508 | . . 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 | 8, 9 | opelxpd 5724 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) | 
| 22 | catcco.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
| 23 | ovex 7464 | . . . . 5 ⊢ (𝑌 Func 𝑍) ∈ V | |
| 24 | ovex 7464 | . . . . 5 ⊢ (𝑋 Func 𝑌) ∈ V | |
| 25 | 23, 24 | mpoex 8104 | . . . 4 ⊢ (𝑔 ∈ (𝑌 Func 𝑍), 𝑓 ∈ (𝑋 Func 𝑌) ↦ (𝑔 ∘func 𝑓)) ∈ V | 
| 26 | 25 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑔 ∈ (𝑌 Func 𝑍), 𝑓 ∈ (𝑋 Func 𝑌) ↦ (𝑔 ∘func 𝑓)) ∈ V) | 
| 27 | 5, 20, 21, 22, 26 | ovmpod 7585 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉 · 𝑍) = (𝑔 ∈ (𝑌 Func 𝑍), 𝑓 ∈ (𝑋 Func 𝑌) ↦ (𝑔 ∘func 𝑓))) | 
| 28 | oveq12 7440 | . . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → (𝑔 ∘func 𝑓) = (𝐺 ∘func 𝐹)) | |
| 29 | 28 | adantl 481 | . 2 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝑔 ∘func 𝑓) = (𝐺 ∘func 𝐹)) | 
| 30 | catcco.g | . 2 ⊢ (𝜑 → 𝐺 ∈ (𝑌 Func 𝑍)) | |
| 31 | catcco.f | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋 Func 𝑌)) | |
| 32 | ovexd 7466 | . 2 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ V) | |
| 33 | 27, 29, 30, 31, 32 | ovmpod 7585 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘func 𝐹)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 × cxp 5683 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 2nd c2nd 8013 Basecbs 17247 compcco 17309 Func cfunc 17899 ∘func ccofu 17901 CatCatccatc 18143 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-hom 17321 df-cco 17322 df-catc 18144 | 
| This theorem is referenced by: catccatid 18151 resscatc 18154 catcisolem 18155 catciso 18156 | 
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