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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 5807 | . . 3 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | simpr 103 | . . 3 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵) | |
3 | 1, 2 | eqeltrrd 2115 | . 2 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝐵) |
4 | df-br 3765 | . 2 ⊢ ((1st ‘𝐴)𝐵(2nd ‘𝐴) ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝐵) | |
5 | 3, 4 | sylibr 137 | 1 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → (1st ‘𝐴)𝐵(2nd ‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1393 〈cop 3378 class class class wbr 3764 Rel wrel 4350 ‘cfv 4902 1st c1st 5765 2nd c2nd 5766 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fv 4910 df-1st 5767 df-2nd 5768 |
This theorem is referenced by: (None) |
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