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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 7105 | . . 3 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | simpr 476 | . . 3 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵) | |
3 | 1, 2 | eqeltrrd 2689 | . 2 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝐵) |
4 | df-br 4584 | . 2 ⊢ ((1st ‘𝐴)𝐵(2nd ‘𝐴) ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝐵) | |
5 | 3, 4 | sylibr 223 | 1 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → (1st ‘𝐴)𝐵(2nd ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 〈cop 4131 class class class wbr 4583 Rel wrel 5043 ‘cfv 5804 1st c1st 7057 2nd c2nd 7058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-1st 7059 df-2nd 7060 |
This theorem is referenced by: cofuval 16365 cofu1 16367 cofu2 16369 cofucl 16371 cofuass 16372 cofulid 16373 cofurid 16374 funcres 16379 cofull 16417 cofth 16418 isnat2 16431 fuccocl 16447 fucidcl 16448 fuclid 16449 fucrid 16450 fucass 16451 fucsect 16455 fucinv 16456 invfuc 16457 fuciso 16458 natpropd 16459 fucpropd 16460 homahom 16512 homadm 16513 homacd 16514 homadmcd 16515 catciso 16580 prfval 16662 prfcl 16666 prf1st 16667 prf2nd 16668 1st2ndprf 16669 evlfcllem 16684 evlfcl 16685 curf1cl 16691 curf2cl 16694 curfcl 16695 uncf1 16699 uncf2 16700 curfuncf 16701 uncfcurf 16702 diag1cl 16705 diag2cl 16709 curf2ndf 16710 yon1cl 16726 oyon1cl 16734 yonedalem1 16735 yonedalem21 16736 yonedalem3a 16737 yonedalem4c 16740 yonedalem22 16741 yonedalem3b 16742 yonedalem3 16743 yonedainv 16744 yonffthlem 16745 yoniso 16748 utop2nei 21864 utop3cls 21865 |
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