Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2736 | . . . . . 6 ⊢ (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴) | |
3 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | 2, 3 | ressval 16735 | . . . . 5 ⊢ ((𝑊 ∈ 𝑈 ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = if((Base‘𝑊) ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉))) |
5 | 1, 4 | sylan 583 | . . . 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 16672 | . . . . . . . . 9 ⊢ Base = Slot 1 | |
8 | wunress.2 | . . . . . . . . . 10 ⊢ (𝜑 → ω ∈ 𝑈) | |
9 | 6, 8 | wunndx 16686 | . . . . . . . . 9 ⊢ (𝜑 → ndx ∈ 𝑈) |
10 | 7, 6, 9 | wunstr 16689 | . . . . . . . 8 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈) |
11 | incom 4101 | . . . . . . . . 9 ⊢ (𝐴 ∩ (Base‘𝑊)) = ((Base‘𝑊) ∩ 𝐴) | |
12 | 7, 6, 1 | wunstr 16689 | . . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑊) ∈ 𝑈) |
13 | 6, 12 | wunin 10292 | . . . . . . . . 9 ⊢ (𝜑 → ((Base‘𝑊) ∩ 𝐴) ∈ 𝑈) |
14 | 11, 13 | eqeltrid 2835 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ (Base‘𝑊)) ∈ 𝑈) |
15 | 6, 10, 14 | wunop 10301 | . . . . . . 7 ⊢ (𝜑 → 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉 ∈ 𝑈) |
16 | 6, 1, 15 | wunsets 16706 | . . . . . 6 ⊢ (𝜑 → (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉) ∈ 𝑈) |
17 | 1, 16 | ifcld 4471 | . . . . 5 ⊢ (𝜑 → if((Base‘𝑊) ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) ∈ 𝑈) |
18 | 17 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → if((Base‘𝑊) ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) ∈ 𝑈) |
19 | 5, 18 | eqeltrd 2831 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) ∈ 𝑈) |
20 | 19 | ex 416 | . 2 ⊢ (𝜑 → (𝐴 ∈ V → (𝑊 ↾s 𝐴) ∈ 𝑈)) |
21 | 6 | wun0 10297 | . . 3 ⊢ (𝜑 → ∅ ∈ 𝑈) |
22 | reldmress 16734 | . . . . 5 ⊢ Rel dom ↾s | |
23 | 22 | ovprc2 7231 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (𝑊 ↾s 𝐴) = ∅) |
24 | 23 | eleq1d 2815 | . . 3 ⊢ (¬ 𝐴 ∈ V → ((𝑊 ↾s 𝐴) ∈ 𝑈 ↔ ∅ ∈ 𝑈)) |
25 | 21, 24 | syl5ibrcom 250 | . 2 ⊢ (𝜑 → (¬ 𝐴 ∈ V → (𝑊 ↾s 𝐴) ∈ 𝑈)) |
26 | 20, 25 | pm2.61d 182 | 1 ⊢ (𝜑 → (𝑊 ↾s 𝐴) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ∩ cin 3852 ⊆ wss 3853 ∅c0 4223 ifcif 4425 〈cop 4533 ‘cfv 6358 (class class class)co 7191 ωcom 7622 WUnicwun 10279 1c1 10695 ndxcnx 16663 sSet csts 16664 Basecbs 16666 ↾s cress 16667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-1cn 10752 ax-addcl 10754 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-oadd 8184 df-omul 8185 df-er 8369 df-ec 8371 df-qs 8375 df-map 8488 df-pm 8489 df-wun 10281 df-ni 10451 df-pli 10452 df-mi 10453 df-lti 10454 df-plpq 10487 df-mpq 10488 df-ltpq 10489 df-enq 10490 df-nq 10491 df-erq 10492 df-plq 10493 df-mq 10494 df-1nq 10495 df-rq 10496 df-ltnq 10497 df-np 10560 df-plp 10562 df-ltp 10564 df-enr 10634 df-nr 10635 df-c 10700 df-nn 11796 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |