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Mirrors > Home > MPE Home > Th. List > 1st2nd | Structured version Visualization version Unicode version |
Description: Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.) |
Ref | Expression |
---|---|
1st2nd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rel 5121 | . . 3 | |
2 | ssel2 3598 | . . 3 | |
3 | 1, 2 | sylanb 489 | . 2 |
4 | 1st2nd2 7205 | . 2 | |
5 | 3, 4 | syl 17 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cvv 3200 wss 3574 cop 4183 cxp 5112 wrel 5119 cfv 5888 c1st 7166 c2nd 7167 |
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-8 1992 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-pow 4843 ax-pr 4906 ax-un 6949 |
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-sbc 3436 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-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fv 5896 df-1st 7168 df-2nd 7169 |
This theorem is referenced by: 2ndrn 7216 1st2ndbr 7217 elopabi 7231 cnvf1olem 7275 ordpinq 9765 addassnq 9780 mulassnq 9781 distrnq 9783 mulidnq 9785 recmulnq 9786 ltexnq 9797 fsumcnv 14504 fprodcnv 14713 cofulid 16550 cofurid 16551 idffth 16593 cofull 16594 cofth 16595 ressffth 16598 isnat2 16608 nat1st2nd 16611 homadmcd 16692 catciso 16757 prf1st 16844 prf2nd 16845 1st2ndprf 16846 curfuncf 16878 uncfcurf 16879 curf2ndf 16887 yonffthlem 16922 yoniso 16925 dprd2dlem2 18439 dprd2dlem1 18440 dprd2da 18441 mdetunilem9 20426 2ndcctbss 21258 utop2nei 22054 utop3cls 22055 caubl 23106 wlkop 26523 nvop2 27463 nvvop 27464 nvop 27531 phop 27673 fgreu 29471 1stpreimas 29483 cvmliftlem1 31267 heiborlem3 33612 rngoi 33698 drngoi 33750 isdrngo1 33755 iscrngo2 33796 |
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