![]() |
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 | β’ (π β π β WUni) |
wunress.2 | β’ (π β Ο β π) |
wunress.3 | β’ (π β π β π) |
Ref | Expression |
---|---|
wunress | β’ (π β (π βΎs π΄) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wunress.3 | . . . . 5 β’ (π β π β π) | |
2 | eqid 2728 | . . . . . 6 β’ (π βΎs π΄) = (π βΎs π΄) | |
3 | eqid 2728 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) | |
4 | 2, 3 | ressval 17219 | . . . . 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 | wunress.2 | . . . . . . . . 9 β’ (π β Ο β π) | |
8 | 6, 7 | basndxelwund 17199 | . . . . . . . 8 β’ (π β (Baseβndx) β π) |
9 | incom 4203 | . . . . . . . . 9 β’ (π΄ β© (Baseβπ)) = ((Baseβπ) β© π΄) | |
10 | baseid 17190 | . . . . . . . . . . 11 β’ Base = Slot (Baseβndx) | |
11 | 10, 6, 1 | wunstr 17164 | . . . . . . . . . 10 β’ (π β (Baseβπ) β π) |
12 | 6, 11 | wunin 10744 | . . . . . . . . 9 β’ (π β ((Baseβπ) β© π΄) β π) |
13 | 9, 12 | eqeltrid 2833 | . . . . . . . 8 β’ (π β (π΄ β© (Baseβπ)) β π) |
14 | 6, 8, 13 | wunop 10753 | . . . . . . 7 β’ (π β β¨(Baseβndx), (π΄ β© (Baseβπ))β© β π) |
15 | 6, 1, 14 | wunsets 17153 | . . . . . 6 β’ (π β (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©) β π) |
16 | 1, 15 | ifcld 4578 | . . . . 5 β’ (π β if((Baseβπ) β π΄, π, (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©)) β π) |
17 | 16 | 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) |
18 | 5, 17 | eqeltrd 2829 | . . 3 β’ ((π β§ π΄ β V) β (π βΎs π΄) β π) |
19 | 18 | ex 411 | . 2 β’ (π β (π΄ β V β (π βΎs π΄) β π)) |
20 | 6 | wun0 10749 | . . 3 β’ (π β β β π) |
21 | reldmress 17218 | . . . . 5 β’ Rel dom βΎs | |
22 | 21 | ovprc2 7466 | . . . 4 β’ (Β¬ π΄ β V β (π βΎs π΄) = β ) |
23 | 22 | eleq1d 2814 | . . 3 β’ (Β¬ π΄ β V β ((π βΎs π΄) β π β β β π)) |
24 | 20, 23 | syl5ibrcom 246 | . 2 β’ (π β (Β¬ π΄ β V β (π βΎs π΄) β π)) |
25 | 19, 24 | 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 1533 β wcel 2098 Vcvv 3473 β© cin 3948 β wss 3949 β c0 4326 ifcif 4532 β¨cop 4638 βcfv 6553 (class class class)co 7426 Οcom 7876 WUnicwun 10731 sSet csts 17139 ndxcnx 17169 Basecbs 17187 βΎs cress 17216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-1cn 11204 ax-addcl 11206 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-oadd 8497 df-omul 8498 df-er 8731 df-ec 8733 df-qs 8737 df-map 8853 df-pm 8854 df-wun 10733 df-ni 10903 df-pli 10904 df-mi 10905 df-lti 10906 df-plpq 10939 df-mpq 10940 df-ltpq 10941 df-enq 10942 df-nq 10943 df-erq 10944 df-plq 10945 df-mq 10946 df-1nq 10947 df-rq 10948 df-ltnq 10949 df-np 11012 df-plp 11014 df-ltp 11016 df-enr 11086 df-nr 11087 df-c 11152 df-nn 12251 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |