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Mirrors > Home > MPE Home > Th. List > wunndx | Structured version Visualization version GIF version |
Description: Closure of the index extractor in an infinite weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Ref | Expression |
---|---|
wunndx.1 | β’ (π β π β WUni) |
wunndx.2 | β’ (π β Ο β π) |
Ref | Expression |
---|---|
wunndx | β’ (π β ndx β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ndx 17071 | . 2 β’ ndx = ( I βΎ β) | |
2 | wunndx.1 | . . 3 β’ (π β π β WUni) | |
3 | wunndx.2 | . . . . 5 β’ (π β Ο β π) | |
4 | 2, 3 | wuncn 11111 | . . . 4 β’ (π β β β π) |
5 | nnsscn 12163 | . . . . 5 β’ β β β | |
6 | 5 | a1i 11 | . . . 4 β’ (π β β β β) |
7 | 2, 4, 6 | wunss 10653 | . . 3 β’ (π β β β π) |
8 | f1oi 6823 | . . . 4 β’ ( I βΎ β):ββ1-1-ontoββ | |
9 | f1of 6785 | . . . 4 β’ (( I βΎ β):ββ1-1-ontoββ β ( I βΎ β):ββΆβ) | |
10 | 8, 9 | mp1i 13 | . . 3 β’ (π β ( I βΎ β):ββΆβ) |
11 | 2, 7, 7, 10 | wunf 10668 | . 2 β’ (π β ( I βΎ β) β π) |
12 | 1, 11 | eqeltrid 2838 | 1 β’ (π β ndx β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2107 β wss 3911 I cid 5531 βΎ cres 5636 βΆwf 6493 β1-1-ontoβwf1o 6496 Οcom 7803 WUnicwun 10641 βcc 11054 βcn 12158 ndxcnx 17070 |
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-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-ndx 17071 |
This theorem is referenced by: basndxelwund 17100 1strwunOLD 17109 wunressOLD 17137 catcoppccl 18008 catcoppcclOLD 18009 catcfuccl 18010 catcfucclOLD 18011 catcxpccl 18100 catcxpcclOLD 18101 |
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