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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 5692 | . . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
| 2 | ssel2 3978 | . . 3 ⊢ ((𝐵 ⊆ (V × V) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (V × V)) | |
| 3 | 1, 2 | sylanb 581 | . 2 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (V × V)) |
| 4 | 1st2nd2 8053 | . 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 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 〈cop 4632 × cxp 5683 Rel wrel 5690 ‘cfv 6561 1st c1st 8012 2nd c2nd 8013 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-1st 8014 df-2nd 8015 |
| This theorem is referenced by: 2ndrn 8066 1st2ndbr 8067 funfv1st2nd 8071 funelss 8072 elopabi 8087 cnvf1olem 8135 ordpinq 10983 addassnq 10998 mulassnq 10999 distrnq 11001 mulidnq 11003 recmulnq 11004 ltexnq 11015 fsumcnv 15809 fprodcnv 16019 cofulid 17935 cofurid 17936 idffth 17980 cofull 17981 cofth 17982 ressffth 17985 isnat2 17996 nat1st2nd 17999 homadmcd 18087 catciso 18156 prf1st 18249 prf2nd 18250 1st2ndprf 18251 curfuncf 18283 uncfcurf 18284 curf2ndf 18292 yonffthlem 18327 yoniso 18330 dprd2dlem2 20060 dprd2dlem1 20061 dprd2da 20062 mdetunilem9 22626 2ndcctbss 23463 utop2nei 24259 utop3cls 24260 caubl 25342 wlkop 29646 nvop2 30627 nvvop 30628 nvop 30695 phop 30837 fgreu 32682 1stpreimas 32715 gsumhashmul 33064 cvmliftlem1 35290 heiborlem3 37820 rngoi 37906 drngoi 37958 isdrngo1 37963 iscrngo2 38004 tposideq 48788 diag1 49004 fuco11bALT 49033 fuco22nat 49041 fucocolem4 49051 precofvalALT 49063 |
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