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Mirrors > Home > MPE Home > Th. List > 1st2nd | Structured version Visualization version GIF 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 | ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rel 5696 | . . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
2 | ssel2 3990 | . . 3 ⊢ ((𝐵 ⊆ (V × V) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (V × V)) | |
3 | 1, 2 | sylanb 581 | . 2 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (V × V)) |
4 | 1st2nd2 8052 | . 2 ⊢ (𝐴 ∈ (V × V) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
5 | 3, 4 | syl 17 | 1 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 〈cop 4637 × cxp 5687 Rel wrel 5694 ‘cfv 6563 1st c1st 8011 2nd c2nd 8012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-1st 8013 df-2nd 8014 |
This theorem is referenced by: 2ndrn 8065 1st2ndbr 8066 funfv1st2nd 8070 funelss 8071 elopabi 8086 cnvf1olem 8134 ordpinq 10981 addassnq 10996 mulassnq 10997 distrnq 10999 mulidnq 11001 recmulnq 11002 ltexnq 11013 fsumcnv 15806 fprodcnv 16016 cofulid 17941 cofurid 17942 idffth 17987 cofull 17988 cofth 17989 ressffth 17992 isnat2 18003 nat1st2nd 18006 homadmcd 18096 catciso 18165 prf1st 18260 prf2nd 18261 1st2ndprf 18262 curfuncf 18295 uncfcurf 18296 curf2ndf 18304 yonffthlem 18339 yoniso 18342 dprd2dlem2 20075 dprd2dlem1 20076 dprd2da 20077 mdetunilem9 22642 2ndcctbss 23479 utop2nei 24275 utop3cls 24276 caubl 25356 wlkop 29661 nvop2 30637 nvvop 30638 nvop 30705 phop 30847 fgreu 32689 1stpreimas 32721 gsumhashmul 33047 cvmliftlem1 35270 heiborlem3 37800 rngoi 37886 drngoi 37938 isdrngo1 37943 iscrngo2 37984 |
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