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Mirrors > Home > MPE Home > Th. List > catcbas | Structured version Visualization version GIF version |
Description: Set of objects of 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) |
Ref | Expression |
---|---|
catcbas | ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | catcbas.c | . . 3 ⊢ 𝐶 = (CatCat‘𝑈) | |
2 | catcbas.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
3 | eqidd 2822 | . . 3 ⊢ (𝜑 → (𝑈 ∩ Cat) = (𝑈 ∩ Cat)) | |
4 | eqidd 2822 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑈 ∩ Cat), 𝑦 ∈ (𝑈 ∩ Cat) ↦ (𝑥 Func 𝑦)) = (𝑥 ∈ (𝑈 ∩ Cat), 𝑦 ∈ (𝑈 ∩ Cat) ↦ (𝑥 Func 𝑦))) | |
5 | eqidd 2822 | . . 3 ⊢ (𝜑 → (𝑣 ∈ ((𝑈 ∩ Cat) × (𝑈 ∩ Cat)), 𝑧 ∈ (𝑈 ∩ Cat) ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))) = (𝑣 ∈ ((𝑈 ∩ Cat) × (𝑈 ∩ Cat)), 𝑧 ∈ (𝑈 ∩ Cat) ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))) | |
6 | 1, 2, 3, 4, 5 | catcval 17346 | . 2 ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), (𝑈 ∩ Cat)〉, 〈(Hom ‘ndx), (𝑥 ∈ (𝑈 ∩ Cat), 𝑦 ∈ (𝑈 ∩ Cat) ↦ (𝑥 Func 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ ((𝑈 ∩ Cat) × (𝑈 ∩ Cat)), 𝑧 ∈ (𝑈 ∩ Cat) ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))〉}) |
7 | catstr 17217 | . 2 ⊢ {〈(Base‘ndx), (𝑈 ∩ Cat)〉, 〈(Hom ‘ndx), (𝑥 ∈ (𝑈 ∩ Cat), 𝑦 ∈ (𝑈 ∩ Cat) ↦ (𝑥 Func 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ ((𝑈 ∩ Cat) × (𝑈 ∩ Cat)), 𝑧 ∈ (𝑈 ∩ Cat) ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))〉} Struct 〈1, ;15〉 | |
8 | baseid 16533 | . 2 ⊢ Base = Slot (Base‘ndx) | |
9 | snsstp1 4743 | . 2 ⊢ {〈(Base‘ndx), (𝑈 ∩ Cat)〉} ⊆ {〈(Base‘ndx), (𝑈 ∩ Cat)〉, 〈(Hom ‘ndx), (𝑥 ∈ (𝑈 ∩ Cat), 𝑦 ∈ (𝑈 ∩ Cat) ↦ (𝑥 Func 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ ((𝑈 ∩ Cat) × (𝑈 ∩ Cat)), 𝑧 ∈ (𝑈 ∩ Cat) ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))〉} | |
10 | inex1g 5215 | . . 3 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Cat) ∈ V) | |
11 | 2, 10 | syl 17 | . 2 ⊢ (𝜑 → (𝑈 ∩ Cat) ∈ V) |
12 | catcbas.b | . 2 ⊢ 𝐵 = (Base‘𝐶) | |
13 | 6, 7, 8, 9, 11, 12 | strfv3 16522 | 1 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3495 ∩ cin 3934 {ctp 4563 〈cop 4565 × cxp 5547 ‘cfv 6349 (class class class)co 7145 ∈ cmpo 7147 2nd c2nd 7679 1c1 10527 5c5 11684 ;cdc 12087 ndxcnx 16470 Basecbs 16473 Hom chom 16566 compcco 16567 Catccat 16925 Func cfunc 17114 ∘func ccofu 17116 CatCatccatc 17344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-oadd 8097 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-z 11971 df-dec 12088 df-uz 12233 df-fz 12883 df-struct 16475 df-ndx 16476 df-slot 16477 df-base 16479 df-hom 16579 df-cco 16580 df-catc 17345 |
This theorem is referenced by: catchomfval 17348 catccofval 17350 catccatid 17352 resscatc 17355 catcisolem 17356 catciso 17357 catcoppccl 17358 catcfuccl 17359 catcxpccl 17447 yoniso 17525 |
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