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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 8064 | . . 3 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
| 2 | simpr 484 | . . 3 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵) | |
| 3 | 1, 2 | eqeltrrd 2842 | . 2 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝐵) |
| 4 | df-br 5144 | . 2 ⊢ ((1st ‘𝐴)𝐵(2nd ‘𝐴) ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝐵) | |
| 5 | 3, 4 | sylibr 234 | 1 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → (1st ‘𝐴)𝐵(2nd ‘𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 〈cop 4632 class class class wbr 5143 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: cofuval 17927 cofu1 17929 cofu2 17931 cofucl 17933 cofuass 17934 cofulid 17935 cofurid 17936 funcres 17941 cofull 17981 cofth 17982 isnat2 17996 fuccocl 18012 fucidcl 18013 fuclid 18014 fucrid 18015 fucass 18016 fucsect 18020 fucinv 18021 invfuc 18022 fuciso 18023 natpropd 18024 fucpropd 18025 homahom 18084 homadm 18085 homacd 18086 homadmcd 18087 catciso 18156 prfval 18244 prfcl 18248 prf1st 18249 prf2nd 18250 1st2ndprf 18251 evlfcllem 18266 evlfcl 18267 curf1cl 18273 curf2cl 18276 curfcl 18277 uncf1 18281 uncf2 18282 curfuncf 18283 uncfcurf 18284 diag1cl 18287 diag2cl 18291 curf2ndf 18292 yon1cl 18308 oyon1cl 18316 yonedalem1 18317 yonedalem21 18318 yonedalem3a 18319 yonedalem4c 18322 yonedalem22 18323 yonedalem3b 18324 yonedalem3 18325 yonedainv 18326 yonffthlem 18327 yoniso 18330 utop2nei 24259 utop3cls 24260 diag1 49004 fuco11b 49032 fuco11bALT 49033 fucocolem1 49048 fucocolem2 49049 fucocolem3 49050 fucocolem4 49051 fucoco 49052 fucorid2 49058 postcofval 49059 postcofcl 49060 precofval 49062 precofvalALT 49063 precofval2 49064 precofcl 49065 thincciso 49102 thincciso2 49104 termc2 49148 |
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