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Mirrors > Home > MPE Home > Th. List > ressbas | Structured version Visualization version Unicode version |
Description: Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) |
Ref | Expression |
---|---|
ressbas.r | ↾s |
ressbas.b |
Ref | Expression |
---|---|
ressbas |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressbas.b | . . . . 5 | |
2 | simp1 1061 | . . . . . 6 | |
3 | sseqin2 3817 | . . . . . 6 | |
4 | 2, 3 | sylib 208 | . . . . 5 |
5 | ressbas.r | . . . . . . 7 ↾s | |
6 | 5, 1 | ressid2 15928 | . . . . . 6 |
7 | 6 | fveq2d 6195 | . . . . 5 |
8 | 1, 4, 7 | 3eqtr4a 2682 | . . . 4 |
9 | 8 | 3expib 1268 | . . 3 |
10 | simp2 1062 | . . . . . 6 | |
11 | fvex 6201 | . . . . . . . 8 | |
12 | 1, 11 | eqeltri 2697 | . . . . . . 7 |
13 | 12 | inex2 4800 | . . . . . 6 |
14 | baseid 15919 | . . . . . . 7 Slot | |
15 | 14 | setsid 15914 | . . . . . 6 sSet |
16 | 10, 13, 15 | sylancl 694 | . . . . 5 sSet |
17 | 5, 1 | ressval2 15929 | . . . . . 6 sSet |
18 | 17 | fveq2d 6195 | . . . . 5 sSet |
19 | 16, 18 | eqtr4d 2659 | . . . 4 |
20 | 19 | 3expib 1268 | . . 3 |
21 | 9, 20 | pm2.61i 176 | . 2 |
22 | 0fv 6227 | . . . . 5 | |
23 | 0ex 4790 | . . . . . 6 | |
24 | 23, 14 | strfvn 15879 | . . . . 5 |
25 | in0 3968 | . . . . 5 | |
26 | 22, 24, 25 | 3eqtr4ri 2655 | . . . 4 |
27 | fvprc 6185 | . . . . . 6 | |
28 | 1, 27 | syl5eq 2668 | . . . . 5 |
29 | 28 | ineq2d 3814 | . . . 4 |
30 | reldmress 15926 | . . . . . . 7 ↾s | |
31 | 30 | ovprc1 6684 | . . . . . 6 ↾s |
32 | 5, 31 | syl5eq 2668 | . . . . 5 |
33 | 32 | fveq2d 6195 | . . . 4 |
34 | 26, 29, 33 | 3eqtr4a 2682 | . . 3 |
35 | 34 | adantr 481 | . 2 |
36 | 21, 35 | pm2.61ian 831 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 cvv 3200 cin 3573 wss 3574 c0 3915 cop 4183 cfv 5888 (class class class)co 6650 cnx 15854 sSet csts 15855 cbs 15857 ↾s cress 15858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rrecex 10008 ax-cnre 10009 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-nn 11021 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 |
This theorem is referenced by: ressbas2 15931 ressbasss 15932 ressress 15938 rescabs 16493 resscatc 16755 resscntz 17764 idrespermg 17831 opprsubg 18636 subrgpropd 18814 sralmod 19187 resstopn 20990 resstps 20991 ressuss 22067 ressxms 22330 ressms 22331 cphsubrglem 22977 resspos 29659 resstos 29660 xrge0base 29685 xrge00 29686 submomnd 29710 suborng 29815 gsumge0cl 40588 sge0tsms 40597 lidlssbas 41922 lidlbas 41923 uzlidlring 41929 dmatALTbas 42190 |
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