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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 17326 | . . . . 5 ⊢ (𝜑 → 𝑈 = (Base‘𝑆)) |
4 | setc1strwun.c | . . . . 5 ⊢ 𝐶 = (Base‘𝑆) | |
5 | 3, 4 | syl6reqr 2872 | . . . 4 ⊢ (𝜑 → 𝐶 = 𝑈) |
6 | 5 | eleq2d 2895 | . . 3 ⊢ (𝜑 → (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ 𝑈)) |
7 | 6 | biimpa 477 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑈) |
8 | eqid 2818 | . . 3 ⊢ {〈(Base‘ndx), 𝑋〉} = {〈(Base‘ndx), 𝑋〉} | |
9 | setc1strwun.o | . . 3 ⊢ (𝜑 → ω ∈ 𝑈) | |
10 | 8, 2, 9 | 1strwun 16589 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → {〈(Base‘ndx), 𝑋〉} ∈ 𝑈) |
11 | 7, 10 | syldan 591 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → {〈(Base‘ndx), 𝑋〉} ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {csn 4557 〈cop 4563 ‘cfv 6348 ωcom 7569 WUnicwun 10110 ndxcnx 16468 Basecbs 16471 SetCatcsetc 17323 |
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 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
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 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-omul 8096 df-er 8278 df-ec 8280 df-qs 8284 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-wun 10112 df-ni 10282 df-pli 10283 df-mi 10284 df-lti 10285 df-plpq 10318 df-mpq 10319 df-ltpq 10320 df-enq 10321 df-nq 10322 df-erq 10323 df-plq 10324 df-mq 10325 df-1nq 10326 df-rq 10327 df-ltnq 10328 df-np 10391 df-plp 10393 df-ltp 10395 df-enr 10465 df-nr 10466 df-c 10531 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-hom 16577 df-cco 16578 df-setc 17324 |
This theorem is referenced by: funcsetcestrclem2 17393 funcsetcestrclem3 17394 funcsetcestrclem7 17399 |
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