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| Description: Obsolete version of wunress 17296 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 17278 | . . . . 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 17249 | . . . . . . . . 9 ⊢ Base = Slot 1 | |
| 8 | wunress.2 | . . . . . . . . . 10 ⊢ (𝜑 → ω ∈ 𝑈) | |
| 9 | 6, 8 | wunndx 17233 | . . . . . . . . 9 ⊢ (𝜑 → ndx ∈ 𝑈) | 
| 10 | 7, 6, 9 | wunstr 17226 | . . . . . . . 8 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈) | 
| 11 | incom 4208 | . . . . . . . . 9 ⊢ (𝐴 ∩ (Base‘𝑊)) = ((Base‘𝑊) ∩ 𝐴) | |
| 12 | baseid 17251 | . . . . . . . . . . 11 ⊢ Base = Slot (Base‘ndx) | |
| 13 | 12, 6, 1 | wunstr 17226 | . . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑊) ∈ 𝑈) | 
| 14 | 6, 13 | wunin 10754 | . . . . . . . . 9 ⊢ (𝜑 → ((Base‘𝑊) ∩ 𝐴) ∈ 𝑈) | 
| 15 | 11, 14 | eqeltrid 2844 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ (Base‘𝑊)) ∈ 𝑈) | 
| 16 | 6, 10, 15 | wunop 10763 | . . . . . . 7 ⊢ (𝜑 → 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉 ∈ 𝑈) | 
| 17 | 6, 1, 16 | wunsets 17215 | . . . . . 6 ⊢ (𝜑 → (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉) ∈ 𝑈) | 
| 18 | 1, 17 | ifcld 4571 | . . . . 5 ⊢ (𝜑 → if((Base‘𝑊) ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) ∈ 𝑈) | 
| 19 | 18 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → if((Base‘𝑊) ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) ∈ 𝑈) | 
| 20 | 5, 19 | eqeltrd 2840 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) ∈ 𝑈) | 
| 21 | 20 | ex 412 | . 2 ⊢ (𝜑 → (𝐴 ∈ V → (𝑊 ↾s 𝐴) ∈ 𝑈)) | 
| 22 | 6 | wun0 10759 | . . 3 ⊢ (𝜑 → ∅ ∈ 𝑈) | 
| 23 | reldmress 17277 | . . . . 5 ⊢ Rel dom ↾s | |
| 24 | 23 | ovprc2 7472 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (𝑊 ↾s 𝐴) = ∅) | 
| 25 | 24 | eleq1d 2825 | . . 3 ⊢ (¬ 𝐴 ∈ V → ((𝑊 ↾s 𝐴) ∈ 𝑈 ↔ ∅ ∈ 𝑈)) | 
| 26 | 22, 25 | syl5ibrcom 247 | . 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 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∩ cin 3949 ⊆ wss 3950 ∅c0 4332 ifcif 4524 〈cop 4631 ‘cfv 6560 (class class class)co 7432 ωcom 7888 WUnicwun 10741 1c1 11157 sSet csts 17201 ndxcnx 17231 Basecbs 17248 ↾s cress 17275 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-1cn 11214 ax-addcl 11216 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-oadd 8511 df-omul 8512 df-er 8746 df-ec 8748 df-qs 8752 df-map 8869 df-pm 8870 df-wun 10743 df-ni 10913 df-pli 10914 df-mi 10915 df-lti 10916 df-plpq 10949 df-mpq 10950 df-ltpq 10951 df-enq 10952 df-nq 10953 df-erq 10954 df-plq 10955 df-mq 10956 df-1nq 10957 df-rq 10958 df-ltnq 10959 df-np 11022 df-plp 11024 df-ltp 11026 df-enr 11096 df-nr 11097 df-c 11162 df-nn 12268 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 | 
| This theorem is referenced by: (None) | 
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