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Mirrors > Home > MPE Home > Th. List > wdomfil | Structured version Visualization version Unicode version |
Description: Weak dominance agrees with normal for finite left sets. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
Ref | Expression |
---|---|
wdomfil | * |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relwdom 8471 | . . . . . . 7 * | |
2 | 1 | brrelex2i 5159 | . . . . . 6 * |
3 | 0domg 8087 | . . . . . 6 | |
4 | 2, 3 | syl 17 | . . . . 5 * |
5 | breq1 4656 | . . . . 5 | |
6 | 4, 5 | syl5ibr 236 | . . . 4 * |
7 | 6 | adantl 482 | . . 3 * |
8 | brwdomn0 8474 | . . . . 5 * | |
9 | 8 | adantl 482 | . . . 4 * |
10 | vex 3203 | . . . . . . . . . 10 | |
11 | fof 6115 | . . . . . . . . . 10 | |
12 | dmfex 7124 | . . . . . . . . . 10 | |
13 | 10, 11, 12 | sylancr 695 | . . . . . . . . 9 |
14 | 13 | adantl 482 | . . . . . . . 8 |
15 | simpl 473 | . . . . . . . 8 | |
16 | simpr 477 | . . . . . . . 8 | |
17 | fodomfi2 8883 | . . . . . . . 8 | |
18 | 14, 15, 16, 17 | syl3anc 1326 | . . . . . . 7 |
19 | 18 | ex 450 | . . . . . 6 |
20 | 19 | adantr 481 | . . . . 5 |
21 | 20 | exlimdv 1861 | . . . 4 |
22 | 9, 21 | sylbid 230 | . . 3 * |
23 | 7, 22 | pm2.61dane 2881 | . 2 * |
24 | domwdom 8479 | . 2 * | |
25 | 23, 24 | impbid1 215 | 1 * |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wex 1704 wcel 1990 wne 2794 cvv 3200 c0 3915 class class class wbr 4653 wf 5884 wfo 5886 cdom 7953 cfn 7955 * cwdom 8462 |
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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
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-rmo 2920 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-int 4476 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-se 5074 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-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-1o 7560 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-wdom 8464 df-card 8765 df-acn 8768 |
This theorem is referenced by: (None) |
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