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Mirrors > Home > MPE Home > Th. List > cnvxp | Structured version Visualization version Unicode version |
Description: The converse of a Cartesian product. Exercise 11 of [Suppes] p. 67. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
cnvxp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvopab 5533 | . . 3 | |
2 | ancom 466 | . . . 4 | |
3 | 2 | opabbii 4717 | . . 3 |
4 | 1, 3 | eqtri 2644 | . 2 |
5 | df-xp 5120 | . . 3 | |
6 | 5 | cnveqi 5297 | . 2 |
7 | df-xp 5120 | . 2 | |
8 | 4, 6, 7 | 3eqtr4i 2654 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wa 384 wceq 1483 wcel 1990 copab 4712 cxp 5112 ccnv 5113 |
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-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 |
This theorem is referenced by: xp0 5552 rnxp 5564 rnxpss 5566 dminxp 5574 imainrect 5575 fparlem3 7279 fparlem4 7280 tposfo 7379 tposf 7380 xpider 7818 xpcomf1o 8049 fpwwe2lem13 9464 trclublem 13734 xpsc 16217 pjdm 20051 tposmap 20263 ordtrest2 21008 ustneism 22027 trust 22033 metustsym 22360 metust 22363 gtiso 29478 padct 29497 ordtcnvNEW 29966 ordtrest2NEW 29969 mbfmcst 30321 eulerpartlemt 30433 0rrv 30513 msrf 31439 mthmpps 31479 elrn3 31652 trclubgNEW 37925 xpexb 38658 |
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