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Mirrors > Home > ILE Home > Th. List > 1st2ndbr | 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 6079 | . . 3 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | simpr 109 | . . 3 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵) | |
3 | 1, 2 | eqeltrrd 2217 | . 2 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝐵) |
4 | df-br 3930 | . 2 ⊢ ((1st ‘𝐴)𝐵(2nd ‘𝐴) ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝐵) | |
5 | 3, 4 | sylibr 133 | 1 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → (1st ‘𝐴)𝐵(2nd ‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 〈cop 3530 class class class wbr 3929 Rel wrel 4544 ‘cfv 5123 1st c1st 6036 2nd c2nd 6037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fv 5131 df-1st 6038 df-2nd 6039 |
This theorem is referenced by: (None) |
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