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Mirrors > Home > ILE Home > Th. List > ecopovsym | Unicode version |
Description: Assuming the operation is commutative, show that the relation , specified by the first hypothesis, is symmetric. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
ecopopr.1 | |
ecopopr.com |
Ref | Expression |
---|---|
ecopovsym |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecopopr.1 | . . . . 5 | |
2 | opabssxp 4613 | . . . . 5 | |
3 | 1, 2 | eqsstri 3129 | . . . 4 |
4 | 3 | brel 4591 | . . 3 |
5 | eqid 2139 | . . . 4 | |
6 | breq1 3932 | . . . . 5 | |
7 | breq2 3933 | . . . . 5 | |
8 | 6, 7 | bibi12d 234 | . . . 4 |
9 | breq2 3933 | . . . . 5 | |
10 | breq1 3932 | . . . . 5 | |
11 | 9, 10 | bibi12d 234 | . . . 4 |
12 | 1 | ecopoveq 6524 | . . . . . 6 |
13 | vex 2689 | . . . . . . . . 9 | |
14 | vex 2689 | . . . . . . . . 9 | |
15 | ecopopr.com | . . . . . . . . 9 | |
16 | 13, 14, 15 | caovcom 5928 | . . . . . . . 8 |
17 | vex 2689 | . . . . . . . . 9 | |
18 | vex 2689 | . . . . . . . . 9 | |
19 | 17, 18, 15 | caovcom 5928 | . . . . . . . 8 |
20 | 16, 19 | eqeq12i 2153 | . . . . . . 7 |
21 | eqcom 2141 | . . . . . . 7 | |
22 | 20, 21 | bitri 183 | . . . . . 6 |
23 | 12, 22 | syl6bb 195 | . . . . 5 |
24 | 1 | ecopoveq 6524 | . . . . . 6 |
25 | 24 | ancoms 266 | . . . . 5 |
26 | 23, 25 | bitr4d 190 | . . . 4 |
27 | 5, 8, 11, 26 | 2optocl 4616 | . . 3 |
28 | 4, 27 | syl 14 | . 2 |
29 | 28 | ibi 175 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 cop 3530 class class class wbr 3929 copab 3988 cxp 4537 (class class class)co 5774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-iota 5088 df-fv 5131 df-ov 5777 |
This theorem is referenced by: ecopover 6527 |
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