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Mirrors > Home > ILE Home > Th. List > cnvexg | Unicode version |
Description: The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.) |
Ref | Expression |
---|---|
cnvexg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 4976 | . . 3 | |
2 | relssdmrn 5118 | . . 3 | |
3 | 1, 2 | ax-mp 5 | . 2 |
4 | df-rn 4609 | . . . 4 | |
5 | rnexg 4863 | . . . 4 | |
6 | 4, 5 | eqeltrrid 2252 | . . 3 |
7 | dfdm4 4790 | . . . 4 | |
8 | dmexg 4862 | . . . 4 | |
9 | 7, 8 | eqeltrrid 2252 | . . 3 |
10 | xpexg 4712 | . . 3 | |
11 | 6, 9, 10 | syl2anc 409 | . 2 |
12 | ssexg 4115 | . 2 | |
13 | 3, 11, 12 | sylancr 411 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 2135 cvv 2721 wss 3111 cxp 4596 ccnv 4597 cdm 4598 crn 4599 wrel 4603 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 df-cnv 4606 df-dm 4608 df-rn 4609 |
This theorem is referenced by: cnvex 5136 relcnvexb 5137 cofunex2g 6072 cnvf1o 6184 brtpos2 6210 tposexg 6217 cnven 6765 cnvct 6766 fopwdom 6793 relcnvfi 6897 ennnfonelemim 12300 pw1nct 13724 |
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