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Mirrors > Home > MPE Home > Th. List > wunressOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of wunress 17199 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 | β’ (π β π β WUni) |
wunress.2 | β’ (π β Ο β π) |
wunress.3 | β’ (π β π β π) |
Ref | Expression |
---|---|
wunressOLD | β’ (π β (π βΎs π΄) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wunress.3 | . . . . 5 β’ (π β π β π) | |
2 | eqid 2730 | . . . . . 6 β’ (π βΎs π΄) = (π βΎs π΄) | |
3 | eqid 2730 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) | |
4 | 2, 3 | ressval 17180 | . . . . 5 β’ ((π β π β§ π΄ β V) β (π βΎs π΄) = if((Baseβπ) β π΄, π, (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©))) |
5 | 1, 4 | sylan 578 | . . . 4 β’ ((π β§ π΄ β V) β (π βΎs π΄) = if((Baseβπ) β π΄, π, (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©))) |
6 | wunress.1 | . . . . . . 7 β’ (π β π β WUni) | |
7 | df-base 17149 | . . . . . . . . 9 β’ Base = Slot 1 | |
8 | wunress.2 | . . . . . . . . . 10 β’ (π β Ο β π) | |
9 | 6, 8 | wunndx 17132 | . . . . . . . . 9 β’ (π β ndx β π) |
10 | 7, 6, 9 | wunstr 17125 | . . . . . . . 8 β’ (π β (Baseβndx) β π) |
11 | incom 4200 | . . . . . . . . 9 β’ (π΄ β© (Baseβπ)) = ((Baseβπ) β© π΄) | |
12 | baseid 17151 | . . . . . . . . . . 11 β’ Base = Slot (Baseβndx) | |
13 | 12, 6, 1 | wunstr 17125 | . . . . . . . . . 10 β’ (π β (Baseβπ) β π) |
14 | 6, 13 | wunin 10710 | . . . . . . . . 9 β’ (π β ((Baseβπ) β© π΄) β π) |
15 | 11, 14 | eqeltrid 2835 | . . . . . . . 8 β’ (π β (π΄ β© (Baseβπ)) β π) |
16 | 6, 10, 15 | wunop 10719 | . . . . . . 7 β’ (π β β¨(Baseβndx), (π΄ β© (Baseβπ))β© β π) |
17 | 6, 1, 16 | wunsets 17114 | . . . . . 6 β’ (π β (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©) β π) |
18 | 1, 17 | ifcld 4573 | . . . . 5 β’ (π β if((Baseβπ) β π΄, π, (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©)) β π) |
19 | 18 | adantr 479 | . . . 4 β’ ((π β§ π΄ β V) β if((Baseβπ) β π΄, π, (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©)) β π) |
20 | 5, 19 | eqeltrd 2831 | . . 3 β’ ((π β§ π΄ β V) β (π βΎs π΄) β π) |
21 | 20 | ex 411 | . 2 β’ (π β (π΄ β V β (π βΎs π΄) β π)) |
22 | 6 | wun0 10715 | . . 3 β’ (π β β β π) |
23 | reldmress 17179 | . . . . 5 β’ Rel dom βΎs | |
24 | 23 | ovprc2 7451 | . . . 4 β’ (Β¬ π΄ β V β (π βΎs π΄) = β ) |
25 | 24 | eleq1d 2816 | . . 3 β’ (Β¬ π΄ β V β ((π βΎs π΄) β π β β β π)) |
26 | 22, 25 | syl5ibrcom 246 | . 2 β’ (π β (Β¬ π΄ β V β (π βΎs π΄) β π)) |
27 | 21, 26 | pm2.61d 179 | 1 β’ (π β (π βΎs π΄) β π) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 = wceq 1539 β wcel 2104 Vcvv 3472 β© cin 3946 β wss 3947 β c0 4321 ifcif 4527 β¨cop 4633 βcfv 6542 (class class class)co 7411 Οcom 7857 WUnicwun 10697 1c1 11113 sSet csts 17100 ndxcnx 17130 Basecbs 17148 βΎs cress 17177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-1cn 11170 ax-addcl 11172 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-oadd 8472 df-omul 8473 df-er 8705 df-ec 8707 df-qs 8711 df-map 8824 df-pm 8825 df-wun 10699 df-ni 10869 df-pli 10870 df-mi 10871 df-lti 10872 df-plpq 10905 df-mpq 10906 df-ltpq 10907 df-enq 10908 df-nq 10909 df-erq 10910 df-plq 10911 df-mq 10912 df-1nq 10913 df-rq 10914 df-ltnq 10915 df-np 10978 df-plp 10980 df-ltp 10982 df-enr 11052 df-nr 11053 df-c 11118 df-nn 12217 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 |
This theorem is referenced by: (None) |
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