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Mirrors > Home > MPE Home > Th. List > wunressOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of wunress 17131 as of 28-Oct-2024. Closure of structure restriction in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
wunress.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wunress.2 | ⊢ (𝜑 → ω ∈ 𝑈) |
wunress.3 | ⊢ (𝜑 → 𝑊 ∈ 𝑈) |
Ref | Expression |
---|---|
wunressOLD | ⊢ (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 17115 | . . . . 5 ⊢ ((𝑊 ∈ 𝑈 ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = if((Base‘𝑊) ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉))) |
5 | 1, 4 | sylan 580 | . . . 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 17084 | . . . . . . . . 9 ⊢ Base = Slot 1 | |
8 | wunress.2 | . . . . . . . . . 10 ⊢ (𝜑 → ω ∈ 𝑈) | |
9 | 6, 8 | wunndx 17067 | . . . . . . . . 9 ⊢ (𝜑 → ndx ∈ 𝑈) |
10 | 7, 6, 9 | wunstr 17060 | . . . . . . . 8 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈) |
11 | incom 4161 | . . . . . . . . 9 ⊢ (𝐴 ∩ (Base‘𝑊)) = ((Base‘𝑊) ∩ 𝐴) | |
12 | baseid 17086 | . . . . . . . . . . 11 ⊢ Base = Slot (Base‘ndx) | |
13 | 12, 6, 1 | wunstr 17060 | . . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑊) ∈ 𝑈) |
14 | 6, 13 | wunin 10649 | . . . . . . . . 9 ⊢ (𝜑 → ((Base‘𝑊) ∩ 𝐴) ∈ 𝑈) |
15 | 11, 14 | eqeltrid 2842 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ (Base‘𝑊)) ∈ 𝑈) |
16 | 6, 10, 15 | wunop 10658 | . . . . . . 7 ⊢ (𝜑 → 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉 ∈ 𝑈) |
17 | 6, 1, 16 | wunsets 17049 | . . . . . 6 ⊢ (𝜑 → (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉) ∈ 𝑈) |
18 | 1, 17 | ifcld 4532 | . . . . 5 ⊢ (𝜑 → if((Base‘𝑊) ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) ∈ 𝑈) |
19 | 18 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → if((Base‘𝑊) ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) ∈ 𝑈) |
20 | 5, 19 | eqeltrd 2838 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) ∈ 𝑈) |
21 | 20 | ex 413 | . 2 ⊢ (𝜑 → (𝐴 ∈ V → (𝑊 ↾s 𝐴) ∈ 𝑈)) |
22 | 6 | wun0 10654 | . . 3 ⊢ (𝜑 → ∅ ∈ 𝑈) |
23 | reldmress 17114 | . . . . 5 ⊢ Rel dom ↾s | |
24 | 23 | ovprc2 7397 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (𝑊 ↾s 𝐴) = ∅) |
25 | 24 | eleq1d 2822 | . . 3 ⊢ (¬ 𝐴 ∈ V → ((𝑊 ↾s 𝐴) ∈ 𝑈 ↔ ∅ ∈ 𝑈)) |
26 | 22, 25 | syl5ibrcom 246 | . 2 ⊢ (𝜑 → (¬ 𝐴 ∈ V → (𝑊 ↾s 𝐴) ∈ 𝑈)) |
27 | 21, 26 | pm2.61d 179 | 1 ⊢ (𝜑 → (𝑊 ↾s 𝐴) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ∩ cin 3909 ⊆ wss 3910 ∅c0 4282 ifcif 4486 〈cop 4592 ‘cfv 6496 (class class class)co 7357 ωcom 7802 WUnicwun 10636 1c1 11052 sSet csts 17035 ndxcnx 17065 Basecbs 17083 ↾s cress 17112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-1cn 11109 ax-addcl 11111 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-oadd 8416 df-omul 8417 df-er 8648 df-ec 8650 df-qs 8654 df-map 8767 df-pm 8768 df-wun 10638 df-ni 10808 df-pli 10809 df-mi 10810 df-lti 10811 df-plpq 10844 df-mpq 10845 df-ltpq 10846 df-enq 10847 df-nq 10848 df-erq 10849 df-plq 10850 df-mq 10851 df-1nq 10852 df-rq 10853 df-ltnq 10854 df-np 10917 df-plp 10919 df-ltp 10921 df-enr 10991 df-nr 10992 df-c 11057 df-nn 12154 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 |
This theorem is referenced by: (None) |
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