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Mirrors > Home > MPE Home > Th. List > catchomfval | Structured version Visualization version GIF version |
Description: Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
catcbas.c | ⊢ 𝐶 = (CatCat‘𝑈) |
catcbas.b | ⊢ 𝐵 = (Base‘𝐶) |
catcbas.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
catchomfval.h | ⊢ 𝐻 = (Hom ‘𝐶) |
Ref | Expression |
---|---|
catchomfval | ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | catchomfval.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
2 | catcbas.c | . . . . 5 ⊢ 𝐶 = (CatCat‘𝑈) | |
3 | catcbas.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
4 | catcbas.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
5 | 2, 4, 3 | catcbas 16954 | . . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat)) |
6 | eqidd 2772 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦))) | |
7 | eqidd 2772 | . . . . 5 ⊢ (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 | 2, 3, 5, 6, 7 | catcval 16953 | . . . 4 ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))〉}) |
9 | 8 | fveq2d 6336 | . . 3 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘{〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))〉})) |
10 | 1, 9 | syl5eq 2817 | . 2 ⊢ (𝜑 → 𝐻 = (Hom ‘{〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))〉})) |
11 | fvex 6342 | . . . . 5 ⊢ (Base‘𝐶) ∈ V | |
12 | 4, 11 | eqeltri 2846 | . . . 4 ⊢ 𝐵 ∈ V |
13 | 12, 12 | mpt2ex 7397 | . . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)) ∈ V |
14 | catstr 16824 | . . . 4 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))〉} Struct 〈1, ;15〉 | |
15 | homid 16283 | . . . 4 ⊢ Hom = Slot (Hom ‘ndx) | |
16 | snsstp2 4483 | . . . 4 ⊢ {〈(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦))〉} ⊆ {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))〉} | |
17 | 14, 15, 16 | strfv 16114 | . . 3 ⊢ ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)) ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)) = (Hom ‘{〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))〉})) |
18 | 13, 17 | mp1i 13 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)) = (Hom ‘{〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))〉})) |
19 | 10, 18 | eqtr4d 2808 | 1 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 Vcvv 3351 {ctp 4320 〈cop 4322 × cxp 5247 ‘cfv 6031 (class class class)co 6793 ↦ cmpt2 6795 2nd c2nd 7314 1c1 10139 5c5 11275 ;cdc 11695 ndxcnx 16061 Basecbs 16064 Hom chom 16160 compcco 16161 Func cfunc 16721 ∘func ccofu 16723 CatCatccatc 16951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11495 df-z 11580 df-dec 11696 df-uz 11889 df-fz 12534 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-hom 16174 df-cco 16175 df-catc 16952 |
This theorem is referenced by: catchom 16956 catccofval 16957 |
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