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Mirrors > Home > MPE Home > Th. List > ercnv | Structured version Visualization version Unicode version |
Description: The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
ercnv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | errel 7751 | . 2 | |
2 | relcnv 5503 | . . 3 | |
3 | id 22 | . . . . . 6 | |
4 | 3 | ersymb 7756 | . . . . 5 |
5 | vex 3203 | . . . . . . 7 | |
6 | vex 3203 | . . . . . . 7 | |
7 | 5, 6 | brcnv 5305 | . . . . . 6 |
8 | df-br 4654 | . . . . . 6 | |
9 | 7, 8 | bitr3i 266 | . . . . 5 |
10 | df-br 4654 | . . . . 5 | |
11 | 4, 9, 10 | 3bitr3g 302 | . . . 4 |
12 | 11 | eqrelrdv2 5219 | . . 3 |
13 | 2, 12 | mpanl1 716 | . 2 |
14 | 1, 13 | mpancom 703 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 cop 4183 class class class wbr 4653 ccnv 5113 wrel 5119 wer 7739 |
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-eu 2474 df-mo 2475 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-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-er 7742 |
This theorem is referenced by: errn 7764 |
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