Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > elres | Structured version Visualization version Unicode version |
Description: Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.) |
Ref | Expression |
---|---|
elres |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 5426 | . . . . 5 | |
2 | elrel 5222 | . . . . 5 | |
3 | 1, 2 | mpan 706 | . . . 4 |
4 | eleq1 2689 | . . . . . . . . 9 | |
5 | 4 | biimpd 219 | . . . . . . . 8 |
6 | vex 3203 | . . . . . . . . . . 11 | |
7 | 6 | opelres 5401 | . . . . . . . . . 10 |
8 | 7 | biimpi 206 | . . . . . . . . 9 |
9 | 8 | ancomd 467 | . . . . . . . 8 |
10 | 5, 9 | syl6com 37 | . . . . . . 7 |
11 | 10 | ancld 576 | . . . . . 6 |
12 | an12 838 | . . . . . 6 | |
13 | 11, 12 | syl6ib 241 | . . . . 5 |
14 | 13 | 2eximdv 1848 | . . . 4 |
15 | 3, 14 | mpd 15 | . . 3 |
16 | rexcom4 3225 | . . . 4 | |
17 | df-rex 2918 | . . . . 5 | |
18 | 17 | exbii 1774 | . . . 4 |
19 | excom 2042 | . . . 4 | |
20 | 16, 18, 19 | 3bitri 286 | . . 3 |
21 | 15, 20 | sylibr 224 | . 2 |
22 | 7 | simplbi2com 657 | . . . . . 6 |
23 | 4 | biimprd 238 | . . . . . 6 |
24 | 22, 23 | syl9 77 | . . . . 5 |
25 | 24 | impd 447 | . . . 4 |
26 | 25 | exlimdv 1861 | . . 3 |
27 | 26 | rexlimiv 3027 | . 2 |
28 | 21, 27 | impbii 199 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wa 384 wceq 1483 wex 1704 wcel 1990 wrex 2913 cop 4183 cres 5116 wrel 5119 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 df-xp 5120 df-rel 5121 df-res 5126 |
This theorem is referenced by: elsnres 5436 eldm3 31651 |
Copyright terms: Public domain | W3C validator |