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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 7738 | . . 3 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | simpr 487 | . . 3 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵) | |
3 | 1, 2 | eqeltrrd 2914 | . 2 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝐵) |
4 | df-br 5067 | . 2 ⊢ ((1st ‘𝐴)𝐵(2nd ‘𝐴) ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝐵) | |
5 | 3, 4 | sylibr 236 | 1 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → (1st ‘𝐴)𝐵(2nd ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 〈cop 4573 class class class wbr 5066 Rel wrel 5560 ‘cfv 6355 1st c1st 7687 2nd c2nd 7688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fv 6363 df-1st 7689 df-2nd 7690 |
This theorem is referenced by: cofuval 17152 cofu1 17154 cofu2 17156 cofucl 17158 cofuass 17159 cofulid 17160 cofurid 17161 funcres 17166 cofull 17204 cofth 17205 isnat2 17218 fuccocl 17234 fucidcl 17235 fuclid 17236 fucrid 17237 fucass 17238 fucsect 17242 fucinv 17243 invfuc 17244 fuciso 17245 natpropd 17246 fucpropd 17247 homahom 17299 homadm 17300 homacd 17301 homadmcd 17302 catciso 17367 prfval 17449 prfcl 17453 prf1st 17454 prf2nd 17455 1st2ndprf 17456 evlfcllem 17471 evlfcl 17472 curf1cl 17478 curf2cl 17481 curfcl 17482 uncf1 17486 uncf2 17487 curfuncf 17488 uncfcurf 17489 diag1cl 17492 diag2cl 17496 curf2ndf 17497 yon1cl 17513 oyon1cl 17521 yonedalem1 17522 yonedalem21 17523 yonedalem3a 17524 yonedalem4c 17527 yonedalem22 17528 yonedalem3b 17529 yonedalem3 17530 yonedainv 17531 yonffthlem 17532 yoniso 17535 utop2nei 22859 utop3cls 22860 |
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