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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 5556 | . . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
2 | ssel2 3961 | . . 3 ⊢ ((𝐵 ⊆ (V × V) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (V × V)) | |
3 | 1, 2 | sylanb 583 | . 2 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (V × V)) |
4 | 1st2nd2 7722 | . 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 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 〈cop 4566 × cxp 5547 Rel wrel 5554 ‘cfv 6349 1st c1st 7681 2nd c2nd 7682 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fv 6357 df-1st 7683 df-2nd 7684 |
This theorem is referenced by: 2ndrn 7734 1st2ndbr 7735 funfv1st2nd 7739 funelss 7740 elopabi 7754 cnvf1olem 7799 ordpinq 10359 addassnq 10374 mulassnq 10375 distrnq 10377 mulidnq 10379 recmulnq 10380 ltexnq 10391 fsumcnv 15122 fprodcnv 15331 cofulid 17154 cofurid 17155 idffth 17197 cofull 17198 cofth 17199 ressffth 17202 isnat2 17212 nat1st2nd 17215 homadmcd 17296 catciso 17361 prf1st 17448 prf2nd 17449 1st2ndprf 17450 curfuncf 17482 uncfcurf 17483 curf2ndf 17491 yonffthlem 17526 yoniso 17529 dprd2dlem2 19156 dprd2dlem1 19157 dprd2da 19158 mdetunilem9 21223 2ndcctbss 22057 utop2nei 22853 utop3cls 22854 caubl 23905 wlkop 27403 nvop2 28379 nvvop 28380 nvop 28447 phop 28589 fgreu 30411 1stpreimas 30435 cvmliftlem1 32527 heiborlem3 35085 rngoi 35171 drngoi 35223 isdrngo1 35228 iscrngo2 35269 |
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