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Mirrors > Home > ILE Home > Th. List > 1st2nd2 | Unicode version |
Description: Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.) |
Ref | Expression |
---|---|
1st2nd2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp6 6067 | . 2 | |
2 | 1 | simplbi 272 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cop 3530 cxp 4537 cfv 5123 c1st 6036 c2nd 6037 |
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-iota 5088 df-fun 5125 df-fv 5131 df-1st 6038 df-2nd 6039 |
This theorem is referenced by: xpopth 6074 eqop 6075 2nd1st 6078 1st2nd 6079 xpmapenlem 6743 djuf1olem 6938 dfplpq2 7162 dfmpq2 7163 enqbreq2 7165 enqdc1 7170 preqlu 7280 prop 7283 elnp1st2nd 7284 cauappcvgprlemladd 7466 elreal2 7638 cnref1o 9440 frecuzrdgrrn 10181 frec2uzrdg 10182 frecuzrdgrcl 10183 frecuzrdgsuc 10187 frecuzrdgrclt 10188 frecuzrdgg 10189 frecuzrdgdomlem 10190 frecuzrdgfunlem 10192 frecuzrdgsuctlem 10196 seq3val 10231 seqvalcd 10232 eucalgval 11735 eucalginv 11737 eucalglt 11738 eucalg 11740 sqpweven 11853 2sqpwodd 11854 qnumdenbi 11870 tx1cn 12438 tx2cn 12439 txdis 12446 psmetxrge0 12501 xmetxpbl 12677 |
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