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Mirrors > Home > MPE Home > Th. List > estrcbas | Structured version Visualization version GIF version |
Description: Set of objects of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.) |
Ref | Expression |
---|---|
estrcbas.c | ⊢ 𝐶 = (ExtStrCat‘𝑈) |
estrcbas.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
Ref | Expression |
---|---|
estrcbas | ⊢ (𝜑 → 𝑈 = (Base‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | estrcbas.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
2 | catstr 16816 | . . . 4 ⊢ {〈(Base‘ndx), 𝑈〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))〉, 〈(comp‘ndx), (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))〉} Struct 〈1, ;15〉 | |
3 | baseid 16119 | . . . 4 ⊢ Base = Slot (Base‘ndx) | |
4 | snsstp1 4490 | . . . 4 ⊢ {〈(Base‘ndx), 𝑈〉} ⊆ {〈(Base‘ndx), 𝑈〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))〉, 〈(comp‘ndx), (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))〉} | |
5 | 2, 3, 4 | strfv 16107 | . . 3 ⊢ (𝑈 ∈ 𝑉 → 𝑈 = (Base‘{〈(Base‘ndx), 𝑈〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))〉, 〈(comp‘ndx), (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))〉})) |
6 | 1, 5 | syl 17 | . 2 ⊢ (𝜑 → 𝑈 = (Base‘{〈(Base‘ndx), 𝑈〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))〉, 〈(comp‘ndx), (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))〉})) |
7 | estrcbas.c | . . . 4 ⊢ 𝐶 = (ExtStrCat‘𝑈) | |
8 | eqidd 2759 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) | |
9 | eqidd 2759 | . . . 4 ⊢ (𝜑 → (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))) = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))) | |
10 | 7, 1, 8, 9 | estrcval 16963 | . . 3 ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝑈〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))〉, 〈(comp‘ndx), (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))〉}) |
11 | 10 | fveq2d 6354 | . 2 ⊢ (𝜑 → (Base‘𝐶) = (Base‘{〈(Base‘ndx), 𝑈〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))〉, 〈(comp‘ndx), (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))〉})) |
12 | 6, 11 | eqtr4d 2795 | 1 ⊢ (𝜑 → 𝑈 = (Base‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1630 ∈ wcel 2137 {ctp 4323 〈cop 4325 × cxp 5262 ∘ ccom 5268 ‘cfv 6047 (class class class)co 6811 ↦ cmpt2 6813 1st c1st 7329 2nd c2nd 7330 ↑𝑚 cmap 8021 1c1 10127 5c5 11263 ;cdc 11683 ndxcnx 16054 Basecbs 16057 Hom chom 16152 compcco 16153 ExtStrCatcestrc 16961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1986 ax-6 2052 ax-7 2088 ax-8 2139 ax-9 2146 ax-10 2166 ax-11 2181 ax-12 2194 ax-13 2389 ax-ext 2738 ax-sep 4931 ax-nul 4939 ax-pow 4990 ax-pr 5053 ax-un 7112 ax-cnex 10182 ax-resscn 10183 ax-1cn 10184 ax-icn 10185 ax-addcl 10186 ax-addrcl 10187 ax-mulcl 10188 ax-mulrcl 10189 ax-mulcom 10190 ax-addass 10191 ax-mulass 10192 ax-distr 10193 ax-i2m1 10194 ax-1ne0 10195 ax-1rid 10196 ax-rnegex 10197 ax-rrecex 10198 ax-cnre 10199 ax-pre-lttri 10200 ax-pre-lttrn 10201 ax-pre-ltadd 10202 ax-pre-mulgt0 10203 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2045 df-eu 2609 df-mo 2610 df-clab 2745 df-cleq 2751 df-clel 2754 df-nfc 2889 df-ne 2931 df-nel 3034 df-ral 3053 df-rex 3054 df-reu 3055 df-rab 3057 df-v 3340 df-sbc 3575 df-csb 3673 df-dif 3716 df-un 3718 df-in 3720 df-ss 3727 df-pss 3729 df-nul 4057 df-if 4229 df-pw 4302 df-sn 4320 df-pr 4322 df-tp 4324 df-op 4326 df-uni 4587 df-int 4626 df-iun 4672 df-br 4803 df-opab 4863 df-mpt 4880 df-tr 4903 df-id 5172 df-eprel 5177 df-po 5185 df-so 5186 df-fr 5223 df-we 5225 df-xp 5270 df-rel 5271 df-cnv 5272 df-co 5273 df-dm 5274 df-rn 5275 df-res 5276 df-ima 5277 df-pred 5839 df-ord 5885 df-on 5886 df-lim 5887 df-suc 5888 df-iota 6010 df-fun 6049 df-fn 6050 df-f 6051 df-f1 6052 df-fo 6053 df-f1o 6054 df-fv 6055 df-riota 6772 df-ov 6814 df-oprab 6815 df-mpt2 6816 df-om 7229 df-1st 7331 df-2nd 7332 df-wrecs 7574 df-recs 7635 df-rdg 7673 df-1o 7727 df-oadd 7731 df-er 7909 df-en 8120 df-dom 8121 df-sdom 8122 df-fin 8123 df-pnf 10266 df-mnf 10267 df-xr 10268 df-ltxr 10269 df-le 10270 df-sub 10458 df-neg 10459 df-nn 11211 df-2 11269 df-3 11270 df-4 11271 df-5 11272 df-6 11273 df-7 11274 df-8 11275 df-9 11276 df-n0 11483 df-z 11568 df-dec 11684 df-uz 11878 df-fz 12518 df-struct 16059 df-ndx 16060 df-slot 16061 df-base 16063 df-hom 16166 df-cco 16167 df-estrc 16962 |
This theorem is referenced by: estrcbasbas 16970 estrccatid 16971 estrchomfeqhom 16975 funcestrcsetclem7 16985 funcestrcsetclem8 16986 funcestrcsetclem9 16987 fthestrcsetc 16989 fullestrcsetc 16990 equivestrcsetc 16991 funcsetcestrclem3 16995 rngcbas 42473 rngchomfval 42474 rngccofval 42478 funcrngcsetc 42506 funcrngcsetcALT 42507 ringcbas 42519 ringchomfval 42520 ringccofval 42524 funcringcsetc 42543 |
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