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Mirrors > Home > MPE Home > Th. List > wunress | Structured version Visualization version GIF version |
Description: Closure of structure restriction in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Ref | Expression |
---|---|
wunress.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wunress.2 | ⊢ (𝜑 → ω ∈ 𝑈) |
wunress.3 | ⊢ (𝜑 → 𝑊 ∈ 𝑈) |
Ref | Expression |
---|---|
wunress | ⊢ (𝜑 → (𝑊 ↾s 𝐴) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wunress.3 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝑈) | |
2 | eqid 2821 | . . . . . 6 ⊢ (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴) | |
3 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | 2, 3 | ressval 16545 | . . . . 5 ⊢ ((𝑊 ∈ 𝑈 ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = if((Base‘𝑊) ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉))) |
5 | 1, 4 | sylan 582 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = if((Base‘𝑊) ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉))) |
6 | wunress.1 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
7 | df-base 16483 | . . . . . . . . 9 ⊢ Base = Slot 1 | |
8 | wunress.2 | . . . . . . . . . 10 ⊢ (𝜑 → ω ∈ 𝑈) | |
9 | 6, 8 | wunndx 16498 | . . . . . . . . 9 ⊢ (𝜑 → ndx ∈ 𝑈) |
10 | 7, 6, 9 | wunstr 16501 | . . . . . . . 8 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈) |
11 | incom 4177 | . . . . . . . . 9 ⊢ (𝐴 ∩ (Base‘𝑊)) = ((Base‘𝑊) ∩ 𝐴) | |
12 | 7, 6, 1 | wunstr 16501 | . . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑊) ∈ 𝑈) |
13 | 6, 12 | wunin 10129 | . . . . . . . . 9 ⊢ (𝜑 → ((Base‘𝑊) ∩ 𝐴) ∈ 𝑈) |
14 | 11, 13 | eqeltrid 2917 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ (Base‘𝑊)) ∈ 𝑈) |
15 | 6, 10, 14 | wunop 10138 | . . . . . . 7 ⊢ (𝜑 → 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉 ∈ 𝑈) |
16 | 6, 1, 15 | wunsets 16518 | . . . . . 6 ⊢ (𝜑 → (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉) ∈ 𝑈) |
17 | 1, 16 | ifcld 4511 | . . . . 5 ⊢ (𝜑 → if((Base‘𝑊) ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) ∈ 𝑈) |
18 | 17 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → if((Base‘𝑊) ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) ∈ 𝑈) |
19 | 5, 18 | eqeltrd 2913 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) ∈ 𝑈) |
20 | 19 | ex 415 | . 2 ⊢ (𝜑 → (𝐴 ∈ V → (𝑊 ↾s 𝐴) ∈ 𝑈)) |
21 | 6 | wun0 10134 | . . 3 ⊢ (𝜑 → ∅ ∈ 𝑈) |
22 | reldmress 16544 | . . . . 5 ⊢ Rel dom ↾s | |
23 | 22 | ovprc2 7190 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (𝑊 ↾s 𝐴) = ∅) |
24 | 23 | eleq1d 2897 | . . 3 ⊢ (¬ 𝐴 ∈ V → ((𝑊 ↾s 𝐴) ∈ 𝑈 ↔ ∅ ∈ 𝑈)) |
25 | 21, 24 | syl5ibrcom 249 | . 2 ⊢ (𝜑 → (¬ 𝐴 ∈ V → (𝑊 ↾s 𝐴) ∈ 𝑈)) |
26 | 20, 25 | pm2.61d 181 | 1 ⊢ (𝜑 → (𝑊 ↾s 𝐴) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 ifcif 4466 〈cop 4566 ‘cfv 6349 (class class class)co 7150 ωcom 7574 WUnicwun 10116 1c1 10532 ndxcnx 16474 sSet csts 16475 Basecbs 16477 ↾s cress 16478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-1cn 10589 ax-addcl 10591 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 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 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 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-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-omul 8101 df-er 8283 df-ec 8285 df-qs 8289 df-map 8402 df-pm 8403 df-wun 10118 df-ni 10288 df-pli 10289 df-mi 10290 df-lti 10291 df-plpq 10324 df-mpq 10325 df-ltpq 10326 df-enq 10327 df-nq 10328 df-erq 10329 df-plq 10330 df-mq 10331 df-1nq 10332 df-rq 10333 df-ltnq 10334 df-np 10397 df-plp 10399 df-ltp 10401 df-enr 10471 df-nr 10472 df-c 10537 df-nn 11633 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 |
This theorem is referenced by: (None) |
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