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Mirrors > Home > MPE Home > Th. List > 1st2nd2 | Structured version Visualization version GIF version |
Description: Reconstruction of a member of a Cartesian product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.) |
Ref | Expression |
---|---|
1st2nd2 | ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp6 8013 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) | |
2 | 1 | simplbi 497 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 〈cop 4634 × cxp 5674 ‘cfv 6543 1st c1st 7977 2nd c2nd 7978 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fv 6551 df-1st 7979 df-2nd 7980 |
This theorem is referenced by: 1st2ndb 8019 xpopth 8020 eqop 8021 2nd1st 8028 1st2nd 8029 opiota 8049 fimaproj 8126 disjen 9140 xpmapenlem 9150 mapunen 9152 djulf1o 9913 djurf1o 9914 djur 9920 r0weon 10013 enqbreq2 10921 nqereu 10930 lterpq 10971 elreal2 11133 cnref1o 12976 ruclem6 16185 ruclem8 16187 ruclem9 16188 ruclem12 16191 eucalgval 16526 eucalginv 16528 eucalglt 16529 eucalg 16531 qnumdenbi 16687 isstruct2 17089 xpsff1o 17520 comfffval2 17652 comfeq 17657 idfucl 17838 funcpropd 17860 coapm 18031 xpccatid 18150 1stfcl 18159 2ndfcl 18160 1st2ndprf 18168 xpcpropd 18171 evlfcl 18185 hofcl 18222 hofpropd 18230 yonedalem3 18243 gsum2dlem2 19887 mdetunilem9 22443 tx1cn 23434 tx2cn 23435 txdis 23457 txlly 23461 txnlly 23462 txhaus 23472 txkgen 23477 txconn 23514 utop3cls 24077 ucnima 24107 fmucndlem 24117 psmetxrge0 24140 imasdsf1olem 24200 cnheiborlem 24801 caublcls 25158 bcthlem1 25173 bcthlem2 25174 bcthlem4 25176 bcthlem5 25177 ovolfcl 25316 ovolfioo 25317 ovolficc 25318 ovolficcss 25319 ovolfsval 25320 ovolicc2lem1 25367 ovolicc2lem5 25371 ovolfs2 25421 uniiccdif 25428 uniioovol 25429 uniiccvol 25430 uniioombllem2a 25432 uniioombllem2 25433 uniioombllem3a 25434 uniioombllem3 25435 uniioombllem4 25436 uniioombllem5 25437 uniioombllem6 25438 dyadmbl 25450 fsumvma 27061 opreu2reuALT 32152 ofpreima 32325 ofpreima2 32326 1stmbfm 33725 2ndmbfm 33726 sibfof 33805 oddpwdcv 33820 txsconnlem 34697 mpst123 34997 bj-elid4 36516 bj-elid6 36518 poimirlem4 36959 poimirlem26 36981 poimirlem27 36982 mblfinlem1 36992 mblfinlem2 36993 ftc2nc 37037 heiborlem8 37153 dvhgrp 40445 dvhlveclem 40446 fvovco 44354 dvnprodlem1 45124 volioof 45165 fvvolioof 45167 fvvolicof 45169 etransclem44 45456 ovolval3 45825 ovolval4lem1 45827 ovolval5lem2 45831 ovnovollem1 45834 ovnovollem2 45835 smfpimbor1lem1 45976 rrx2xpref1o 47569 |
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