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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 | 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 2733 | . . . . . 6 β’ (π βΎs π΄) = (π βΎs π΄) | |
3 | eqid 2733 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) | |
4 | 2, 3 | ressval 17120 | . . . . 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 581 | . . . 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 17100 | . . . . . . . 8 β’ (π β (Baseβndx) β π) |
9 | incom 4162 | . . . . . . . . 9 β’ (π΄ β© (Baseβπ)) = ((Baseβπ) β© π΄) | |
10 | baseid 17091 | . . . . . . . . . . 11 β’ Base = Slot (Baseβndx) | |
11 | 10, 6, 1 | wunstr 17065 | . . . . . . . . . 10 β’ (π β (Baseβπ) β π) |
12 | 6, 11 | wunin 10654 | . . . . . . . . 9 β’ (π β ((Baseβπ) β© π΄) β π) |
13 | 9, 12 | eqeltrid 2838 | . . . . . . . 8 β’ (π β (π΄ β© (Baseβπ)) β π) |
14 | 6, 8, 13 | wunop 10663 | . . . . . . 7 β’ (π β β¨(Baseβndx), (π΄ β© (Baseβπ))β© β π) |
15 | 6, 1, 14 | wunsets 17054 | . . . . . 6 β’ (π β (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©) β π) |
16 | 1, 15 | ifcld 4533 | . . . . 5 β’ (π β if((Baseβπ) β π΄, π, (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©)) β π) |
17 | 16 | adantr 482 | . . . 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 2834 | . . 3 β’ ((π β§ π΄ β V) β (π βΎs π΄) β π) |
19 | 18 | ex 414 | . 2 β’ (π β (π΄ β V β (π βΎs π΄) β π)) |
20 | 6 | wun0 10659 | . . 3 β’ (π β β β π) |
21 | reldmress 17119 | . . . . 5 β’ Rel dom βΎs | |
22 | 21 | ovprc2 7398 | . . . 4 β’ (Β¬ π΄ β V β (π βΎs π΄) = β ) |
23 | 22 | eleq1d 2819 | . . 3 β’ (Β¬ π΄ β V β ((π βΎs π΄) β π β β β π)) |
24 | 20, 23 | syl5ibrcom 247 | . 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 397 = wceq 1542 β wcel 2107 Vcvv 3444 β© cin 3910 β wss 3911 β c0 4283 ifcif 4487 β¨cop 4593 βcfv 6497 (class class class)co 7358 Οcom 7803 WUnicwun 10641 sSet csts 17040 ndxcnx 17070 Basecbs 17088 βΎs cress 17117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-1cn 11114 ax-addcl 11116 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-oadd 8417 df-omul 8418 df-er 8651 df-ec 8653 df-qs 8657 df-map 8770 df-pm 8771 df-wun 10643 df-ni 10813 df-pli 10814 df-mi 10815 df-lti 10816 df-plpq 10849 df-mpq 10850 df-ltpq 10851 df-enq 10852 df-nq 10853 df-erq 10854 df-plq 10855 df-mq 10856 df-1nq 10857 df-rq 10858 df-ltnq 10859 df-np 10922 df-plp 10924 df-ltp 10926 df-enr 10996 df-nr 10997 df-c 11062 df-nn 12159 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 |
This theorem is referenced by: (None) |
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