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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.) (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 2735 | . . . . . 6 ⊢ (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴) | |
3 | eqid 2735 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | 2, 3 | ressval 17277 | . . . . 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 | wunress.2 | . . . . . . . . 9 ⊢ (𝜑 → ω ∈ 𝑈) | |
8 | 6, 7 | basndxelwund 17257 | . . . . . . . 8 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈) |
9 | incom 4217 | . . . . . . . . 9 ⊢ (𝐴 ∩ (Base‘𝑊)) = ((Base‘𝑊) ∩ 𝐴) | |
10 | baseid 17248 | . . . . . . . . . . 11 ⊢ Base = Slot (Base‘ndx) | |
11 | 10, 6, 1 | wunstr 17222 | . . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑊) ∈ 𝑈) |
12 | 6, 11 | wunin 10751 | . . . . . . . . 9 ⊢ (𝜑 → ((Base‘𝑊) ∩ 𝐴) ∈ 𝑈) |
13 | 9, 12 | eqeltrid 2843 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ (Base‘𝑊)) ∈ 𝑈) |
14 | 6, 8, 13 | wunop 10760 | . . . . . . 7 ⊢ (𝜑 → 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉 ∈ 𝑈) |
15 | 6, 1, 14 | wunsets 17211 | . . . . . 6 ⊢ (𝜑 → (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉) ∈ 𝑈) |
16 | 1, 15 | ifcld 4577 | . . . . 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 2839 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) ∈ 𝑈) |
19 | 18 | ex 412 | . 2 ⊢ (𝜑 → (𝐴 ∈ V → (𝑊 ↾s 𝐴) ∈ 𝑈)) |
20 | 6 | wun0 10756 | . . 3 ⊢ (𝜑 → ∅ ∈ 𝑈) |
21 | reldmress 17276 | . . . . 5 ⊢ Rel dom ↾s | |
22 | 21 | ovprc2 7471 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (𝑊 ↾s 𝐴) = ∅) |
23 | 22 | eleq1d 2824 | . . 3 ⊢ (¬ 𝐴 ∈ V → ((𝑊 ↾s 𝐴) ∈ 𝑈 ↔ ∅ ∈ 𝑈)) |
24 | 20, 23 | syl5ibrcom 247 | . 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 1537 ∈ wcel 2106 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 ifcif 4531 〈cop 4637 ‘cfv 6563 (class class class)co 7431 ωcom 7887 WUnicwun 10738 sSet csts 17197 ndxcnx 17227 Basecbs 17245 ↾s cress 17274 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-1cn 11211 ax-addcl 11213 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-omul 8510 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-pm 8868 df-wun 10740 df-ni 10910 df-pli 10911 df-mi 10912 df-lti 10913 df-plpq 10946 df-mpq 10947 df-ltpq 10948 df-enq 10949 df-nq 10950 df-erq 10951 df-plq 10952 df-mq 10953 df-1nq 10954 df-rq 10955 df-ltnq 10956 df-np 11019 df-plp 11021 df-ltp 11023 df-enr 11093 df-nr 11094 df-c 11159 df-nn 12265 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 |
This theorem is referenced by: (None) |
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