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Mirrors > Home > MPE Home > Th. List > 1st2ndbr | Structured version Visualization version GIF version |
Description: Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.) |
Ref | Expression |
---|---|
1st2ndbr | ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → (1st ‘𝐴)𝐵(2nd ‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1st2nd 7727 | . . 3 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | simpr 485 | . . 3 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵) | |
3 | 1, 2 | eqeltrrd 2911 | . 2 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝐵) |
4 | df-br 5058 | . 2 ⊢ ((1st ‘𝐴)𝐵(2nd ‘𝐴) ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝐵) | |
5 | 3, 4 | sylibr 235 | 1 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → (1st ‘𝐴)𝐵(2nd ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 〈cop 4563 class class class wbr 5057 Rel wrel 5553 ‘cfv 6348 1st c1st 7676 2nd c2nd 7677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fv 6356 df-1st 7678 df-2nd 7679 |
This theorem is referenced by: cofuval 17140 cofu1 17142 cofu2 17144 cofucl 17146 cofuass 17147 cofulid 17148 cofurid 17149 funcres 17154 cofull 17192 cofth 17193 isnat2 17206 fuccocl 17222 fucidcl 17223 fuclid 17224 fucrid 17225 fucass 17226 fucsect 17230 fucinv 17231 invfuc 17232 fuciso 17233 natpropd 17234 fucpropd 17235 homahom 17287 homadm 17288 homacd 17289 homadmcd 17290 catciso 17355 prfval 17437 prfcl 17441 prf1st 17442 prf2nd 17443 1st2ndprf 17444 evlfcllem 17459 evlfcl 17460 curf1cl 17466 curf2cl 17469 curfcl 17470 uncf1 17474 uncf2 17475 curfuncf 17476 uncfcurf 17477 diag1cl 17480 diag2cl 17484 curf2ndf 17485 yon1cl 17501 oyon1cl 17509 yonedalem1 17510 yonedalem21 17511 yonedalem3a 17512 yonedalem4c 17515 yonedalem22 17516 yonedalem3b 17517 yonedalem3 17518 yonedainv 17519 yonffthlem 17520 yoniso 17523 utop2nei 22786 utop3cls 22787 |
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