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Mirrors > Home > MPE Home > Th. List > setc1strwun | Structured version Visualization version GIF version |
Description: A constructed one-slot structure with the objects of the category of sets as base set in a weak universe. (Contributed by AV, 27-Mar-2020.) |
Ref | Expression |
---|---|
setc1strwun.s | ⊢ 𝑆 = (SetCat‘𝑈) |
setc1strwun.c | ⊢ 𝐶 = (Base‘𝑆) |
setc1strwun.u | ⊢ (𝜑 → 𝑈 ∈ WUni) |
setc1strwun.o | ⊢ (𝜑 → ω ∈ 𝑈) |
Ref | Expression |
---|---|
setc1strwun | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → {〈(Base‘ndx), 𝑋〉} ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setc1strwun.s | . . . . . 6 ⊢ 𝑆 = (SetCat‘𝑈) | |
2 | setc1strwun.u | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
3 | 1, 2 | setcbas 17042 | . . . . 5 ⊢ (𝜑 → 𝑈 = (Base‘𝑆)) |
4 | setc1strwun.c | . . . . 5 ⊢ 𝐶 = (Base‘𝑆) | |
5 | 3, 4 | syl6reqr 2852 | . . . 4 ⊢ (𝜑 → 𝐶 = 𝑈) |
6 | 5 | eleq2d 2864 | . . 3 ⊢ (𝜑 → (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ 𝑈)) |
7 | 6 | biimpa 469 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑈) |
8 | eqid 2799 | . . 3 ⊢ {〈(Base‘ndx), 𝑋〉} = {〈(Base‘ndx), 𝑋〉} | |
9 | setc1strwun.o | . . 3 ⊢ (𝜑 → ω ∈ 𝑈) | |
10 | 8, 2, 9 | 1strwun 16303 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → {〈(Base‘ndx), 𝑋〉} ∈ 𝑈) |
11 | 7, 10 | syldan 586 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → {〈(Base‘ndx), 𝑋〉} ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 {csn 4368 〈cop 4374 ‘cfv 6101 ωcom 7299 WUnicwun 9810 ndxcnx 16181 Basecbs 16184 SetCatcsetc 17039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-omul 7804 df-er 7982 df-ec 7984 df-qs 7988 df-map 8097 df-pm 8098 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-wun 9812 df-ni 9982 df-pli 9983 df-mi 9984 df-lti 9985 df-plpq 10018 df-mpq 10019 df-ltpq 10020 df-enq 10021 df-nq 10022 df-erq 10023 df-plq 10024 df-mq 10025 df-1nq 10026 df-rq 10027 df-ltnq 10028 df-np 10091 df-plp 10093 df-ltp 10095 df-enr 10165 df-nr 10166 df-c 10230 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-fz 12581 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-hom 16291 df-cco 16292 df-setc 17040 |
This theorem is referenced by: funcsetcestrclem2 17110 funcsetcestrclem3 17111 funcsetcestrclem7 17116 |
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