![]() |
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.) (Proof shortened by AV, 28-Oct-2024.) |
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 2731 | . . . . . 6 ⊢ (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴) | |
3 | eqid 2731 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | 2, 3 | ressval 17183 | . . . . 5 ⊢ ((𝑊 ∈ 𝑈 ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = if((Base‘𝑊) ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉))) |
5 | 1, 4 | sylan 579 | . . . 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 | wunress.2 | . . . . . . . . 9 ⊢ (𝜑 → ω ∈ 𝑈) | |
8 | 6, 7 | basndxelwund 17163 | . . . . . . . 8 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈) |
9 | incom 4201 | . . . . . . . . 9 ⊢ (𝐴 ∩ (Base‘𝑊)) = ((Base‘𝑊) ∩ 𝐴) | |
10 | baseid 17154 | . . . . . . . . . . 11 ⊢ Base = Slot (Base‘ndx) | |
11 | 10, 6, 1 | wunstr 17128 | . . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑊) ∈ 𝑈) |
12 | 6, 11 | wunin 10714 | . . . . . . . . 9 ⊢ (𝜑 → ((Base‘𝑊) ∩ 𝐴) ∈ 𝑈) |
13 | 9, 12 | eqeltrid 2836 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ (Base‘𝑊)) ∈ 𝑈) |
14 | 6, 8, 13 | wunop 10723 | . . . . . . 7 ⊢ (𝜑 → 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉 ∈ 𝑈) |
15 | 6, 1, 14 | wunsets 17117 | . . . . . 6 ⊢ (𝜑 → (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉) ∈ 𝑈) |
16 | 1, 15 | ifcld 4574 | . . . . 5 ⊢ (𝜑 → if((Base‘𝑊) ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) ∈ 𝑈) |
17 | 16 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → if((Base‘𝑊) ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) ∈ 𝑈) |
18 | 5, 17 | eqeltrd 2832 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) ∈ 𝑈) |
19 | 18 | ex 412 | . 2 ⊢ (𝜑 → (𝐴 ∈ V → (𝑊 ↾s 𝐴) ∈ 𝑈)) |
20 | 6 | wun0 10719 | . . 3 ⊢ (𝜑 → ∅ ∈ 𝑈) |
21 | reldmress 17182 | . . . . 5 ⊢ Rel dom ↾s | |
22 | 21 | ovprc2 7452 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (𝑊 ↾s 𝐴) = ∅) |
23 | 22 | eleq1d 2817 | . . 3 ⊢ (¬ 𝐴 ∈ V → ((𝑊 ↾s 𝐴) ∈ 𝑈 ↔ ∅ ∈ 𝑈)) |
24 | 20, 23 | syl5ibrcom 246 | . 2 ⊢ (𝜑 → (¬ 𝐴 ∈ V → (𝑊 ↾s 𝐴) ∈ 𝑈)) |
25 | 19, 24 | pm2.61d 179 | 1 ⊢ (𝜑 → (𝑊 ↾s 𝐴) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 Vcvv 3473 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 ifcif 4528 〈cop 4634 ‘cfv 6543 (class class class)co 7412 ωcom 7859 WUnicwun 10701 sSet csts 17103 ndxcnx 17133 Basecbs 17151 ↾s cress 17180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-1cn 11174 ax-addcl 11176 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-oadd 8476 df-omul 8477 df-er 8709 df-ec 8711 df-qs 8715 df-map 8828 df-pm 8829 df-wun 10703 df-ni 10873 df-pli 10874 df-mi 10875 df-lti 10876 df-plpq 10909 df-mpq 10910 df-ltpq 10911 df-enq 10912 df-nq 10913 df-erq 10914 df-plq 10915 df-mq 10916 df-1nq 10917 df-rq 10918 df-ltnq 10919 df-np 10982 df-plp 10984 df-ltp 10986 df-enr 11056 df-nr 11057 df-c 11122 df-nn 12220 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |