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Mirrors > Home > ILE Home > Th. List > relcnvfi | Unicode version |
Description: If a relation is finite, its converse is as well. (Contributed by Jim Kingdon, 5-Feb-2022.) |
Ref | Expression |
---|---|
relcnvfi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrel2 4989 | . . . . 5 | |
2 | 1 | biimpi 119 | . . . 4 |
3 | 2 | adantr 274 | . . 3 |
4 | simpr 109 | . . 3 | |
5 | 3, 4 | eqeltrd 2216 | . 2 |
6 | relcnv 4917 | . . . 4 | |
7 | cnvexg 5076 | . . . 4 | |
8 | cnven 6702 | . . . 4 | |
9 | 6, 7, 8 | sylancr 410 | . . 3 |
10 | 9 | adantl 275 | . 2 |
11 | enfii 6768 | . 2 | |
12 | 5, 10, 11 | syl2anc 408 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cvv 2686 class class class wbr 3929 ccnv 4538 wrel 4544 cen 6632 cfn 6634 |
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-13 1491 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 ax-un 4355 |
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-sbc 2910 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-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1st 6038 df-2nd 6039 df-er 6429 df-en 6635 df-fin 6637 |
This theorem is referenced by: funrnfi 6830 fsumcnv 11206 |
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