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Mirrors > Home > MPE Home > Th. List > catccofval | 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) |
Ref | Expression |
---|---|
catccofval | ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | catcbas.c | . . . 4 ⊢ 𝐶 = (CatCat‘𝑈) | |
2 | catcbas.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
3 | catcbas.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
4 | 1, 3, 2 | catcbas 17359 | . . . 4 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat)) |
5 | eqid 2823 | . . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
6 | 1, 3, 2, 5 | catchomfval 17360 | . . . 4 ⊢ (𝜑 → (Hom ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦))) |
7 | eqidd 2824 | . . . 4 ⊢ (𝜑 → (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))) = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))) | |
8 | 1, 2, 4, 6, 7 | catcval 17358 | . . 3 ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (Hom ‘𝐶)〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))〉}) |
9 | 8 | fveq2d 6676 | . 2 ⊢ (𝜑 → (comp‘𝐶) = (comp‘{〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (Hom ‘𝐶)〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))〉})) |
10 | catcco.o | . 2 ⊢ · = (comp‘𝐶) | |
11 | 3 | fvexi 6686 | . . . . 5 ⊢ 𝐵 ∈ V |
12 | 11, 11 | xpex 7478 | . . . 4 ⊢ (𝐵 × 𝐵) ∈ V |
13 | 12, 11 | mpoex 7779 | . . 3 ⊢ (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))) ∈ V |
14 | catstr 17229 | . . . 4 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (Hom ‘𝐶)〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))〉} Struct 〈1, ;15〉 | |
15 | ccoid 16692 | . . . 4 ⊢ comp = Slot (comp‘ndx) | |
16 | snsstp3 4753 | . . . 4 ⊢ {〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))〉} ⊆ {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (Hom ‘𝐶)〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))〉} | |
17 | 14, 15, 16 | strfv 16533 | . . 3 ⊢ ((𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))) ∈ V → (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))) = (comp‘{〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (Hom ‘𝐶)〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))〉})) |
18 | 13, 17 | ax-mp 5 | . 2 ⊢ (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))) = (comp‘{〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (Hom ‘𝐶)〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))〉}) |
19 | 9, 10, 18 | 3eqtr4g 2883 | 1 ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 {ctp 4573 〈cop 4575 × cxp 5555 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 2nd c2nd 7690 1c1 10540 5c5 11698 ;cdc 12101 ndxcnx 16482 Basecbs 16485 Hom chom 16578 compcco 16579 Func cfunc 17126 ∘func ccofu 17128 CatCatccatc 17356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-hom 16591 df-cco 16592 df-catc 17357 |
This theorem is referenced by: catcco 17363 |
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